**Cite This:**

*J. Chem. Theory Comput.*2021, 17, 5, 2886-2905

# Linear-Scaling Open-Shell MP2 Approach: Algorithm, Benchmarks, and Large-Scale ApplicationsClick to copy article linkArticle link copied!

- P. Bernát SzabóP. Bernát SzabóDepartment of Physical Chemistry and Materials Science, Budapest University of Technology and Economics, P.O. Box 91, H-1521 Budapest, HungaryMore by P. Bernát Szabó
- József CsókaJózsef CsókaDepartment of Physical Chemistry and Materials Science, Budapest University of Technology and Economics, P.O. Box 91, H-1521 Budapest, HungaryMore by József Csóka
- Mihály Kállay
*****Mihály KállayDepartment of Physical Chemistry and Materials Science, Budapest University of Technology and Economics, P.O. Box 91, H-1521 Budapest, Hungary*****Email: [email protected]More by Mihály Kállay - Péter R. Nagy
*****Péter R. Nagy*****Email: [email protected]More by Péter R. Nagy

## Abstract

A linear-scaling local second-order Møller–Plesset (MP2) method is presented for high-spin open-shell molecules based on restricted open-shell (RO) reference functions. The open-shell local MP2 (LMP2) approach inherits the iteration- and redundancy-free formulation and the completely integral-direct, OpenMP-parallel, and memory and disk use economic algorithms of our closed-shell LMP2 implementation. By utilizing restricted local molecular orbitals for the demanding integral transformation step and by introducing a novel long-range spin-polarization approximation, the computational cost of RO-LMP2 approaches that of closed-shell LMP2. Extensive benchmarks were performed for reactions of radicals, ionization potentials, as well as spin-state splittings of carbenes and transition-metal complexes. Compared to the conventional MP2 reference for systems of up to 175 atoms, local errors of at most 0.1 kcal/mol were found, which are well below the intrinsic accuracy of MP2. RO-LMP2 computations are presented for challenging protein models of up to 601 atoms and 11 000 basis functions, which involve either spin states of a complexed iron ion or a highly delocalized singly occupied orbital. The corresponding runtimes of 9–15 h obtained with a single, many-core CPU demonstrate that MP2, as well as spin-scaled MP2 and double-hybrid density functional methods, become widely accessible for open-shell systems of unprecedented size and complexity.

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### License Summary*

You are free to share(copy and redistribute) this article in any medium or format and to adapt(remix, transform, and build upon) the material for any purpose, even commercially within the parameters below:

Creative Commons (CC): This is a Creative Commons license.

Attribution (BY): Credit must be given to the creator.

*Disclaimer

This summary highlights only some of the key features and terms of the actual license. It is not a license and has no legal value. Carefully review the actual license before using these materials.

### License Summary*

You are free to share(copy and redistribute) this article in any medium or format and to adapt(remix, transform, and build upon) the material for any purpose, even commercially within the parameters below:

Creative Commons (CC): This is a Creative Commons license.

Attribution (BY): Credit must be given to the creator.

*Disclaimer

This summary highlights only some of the key features and terms of the actual license. It is not a license and has no legal value. Carefully review the actual license before using these materials.

## 1. Introduction

*n*-electron valence state perturbation theory (NEVPT2) (99) for the PNO generation, (63) while the PNO methods of Werner and Ma utilize a spin-adapted MP2 formulation (PNO-RMP2). (65) Both approaches share the benefit of spin-free amplitudes useful to obtain a spin-restricted set of PNOs at the price of a somewhat more complicated second-order treatment.

## 2. Theoretical Background

*i*,

*j*,

*k*, ...,

*I*,

*J*,

*K*, ...) and (

*a*,

*b*,

*c*, ...,

*A*,

*B*,

*C*, ...) indices for the occupied and virtual subsets, respectively. Lower (upper) case indices label the spin-up (spin-down) set of semicanonical orbitals. Local approximations will rely on localized molecular orbitals (LMOs) obtained from a restricted open-shell reference , while these LMOs will be labeled as

*i*′,

*j*′,

*k*′, ... (

*I*′,

*J*′,

*K*′, ...), respectively, when occupied by spin-up (spin-down) electrons.

i, j, k, ... | (semi-)canonical occupied orbitals (spin-up) |

I, J, K, ... | (semi-)canonical occupied orbitals (spin-down) |

a, b, c, ... | (semi-)canonical virtual orbitals (spin-up) |

A, B, C, ... | (semi-)canonical virtual orbitals (spin-down) |

i′, j′, k′, ... | localized restricted occupied orbitals (spin-up) |

I′, J′, K′, ... | localized restricted occupied orbitals (spin-down) |

localized restricted occupied orbitals (spatial) | |

ĩ, ..., ã, ... | (semi-)canonical orbitals in the primary or extended domain (spin-up) |

Ĩ, ..., Ã, ... | (semi-)canonical orbitals in the primary or extended domain (spin-down) |

μ, ν, λ, ... | atomic orbitals |

P, Q, ... | auxiliary functions for the DF approximation |

### 2.1. Canonical Open-Shell MP2 Ansatz

*E*

_{MP2}

^{c}) is calculated relying on an unrestricted formalism

*t*

_{i}

^{a}and

*t*

_{ij}

^{ab}... denote MP1 amplitudes corresponding to single and double excitations. Moreover,

*f*

_{i}

^{a}and ⟨

*ab*|

*ij*⟩ stand for Fock-matrix elements and electron repulsion integrals (ERIs) in the Dirac notation, respectively, while ⟨

*ab*∥

*ij*⟩ = ⟨

*ab*|

*ij*⟩ – ⟨

*ab*|

*ji*⟩. The beneficial properties of this correlation energy expression include the invariance to the separate unitary transformation of the occupied and unoccupied MOs. This opens the possibility of introducing local correlation approximations exploiting LMOs. Naturally, eq 1 is equivalent to the closed-shell MP2 correlation energy in the special case of closed-shell systems.

### 2.2. Open-Shell Local MP2 Ansatz

*E*

_{MP2}

^{c}can be rewritten in terms of restricted orbitals. Then,

*E*

_{MP2}

^{c}is expressed in terms of correlation energy contributions of occupied orbitals by separating one occupied index in the summations of eq 1

*i*′ (

*I*′) refers to orbitals with the same spatial component as but occupied by at most one spin-up (spin-down) electron. Here, we also assume that the restricted orbitals of eq 2 are LMOs; hence, the orbital indices are primed. The equivalence of eqs 1 and 2 can be utilized to define the correlation energy contributions of individual LMOs occupied by spin-up and spin-down electrons

*i*→

*i*′ (

*I*→

*I*′) and the restriction of the summations over this index, the complete permutational symmetry of the MP1 amplitudes and two-electron integrals is lost. Consequently, the final terms of eqs 3 and 4 cannot be combined into a single term like in the conventional theory, which explains the appearance of the additional, sixth type of term (third one of eq 4). Note also that the δ

*E*

_{I′}contribution of a singly occupied (SO) restricted LMO is zero by definition; therefore, the correlation energy contribution of the SO LMOs contains only half as many terms as for a doubly occupied (DO) LMO.

## 3. Local MP2 Algorithm

### 3.1. Self-Consistent Field Calculation

*S*(

*S*+ 1) eigenvalue for the square of the spin operator,

*Ŝ*

^{2}, with

*S*as the spin quantum number. To circumvent this, the QROs are constructed as the eigenvectors of the total density matrix of the unrestricted computation. (102) The orbitals obtained in this way with occupation numbers close to 2, 1, and 0 are selected to be DO, SO, and unoccupied in the QRO determinant, respectively, which becomes an eigenfunction of

*Ŝ*

^{2}by construction. Our numerical experience to date is in line with the findings of Neese and co-workers (63,102) that the QRO determinant provides a reliable reference (when the RO solution is unavailable) with a somewhat higher energy than the corresponding ROHF/ROKS determinant.

### 3.2. Orbital Localization

### 3.3. PAO Construction

*a*

_{μ}⟩ denotes the PAO projected from AO |μ⟩, and the summation runs over both DO and SO MOs. The PAOs of eq 6 span only the virtual subspace of spin-up electrons because the projection makes the PAOs orthogonal also to the SO subspace. Therefore, the unoccupied subspace of the spin-down electrons is spanned by the union of the above PAOs and all SOMOs. In other words, the SOMOs have a dual role: they are occupied in the spin-up and unoccupied in the spin-down orbital set.

### 3.4. Pair Energy Calculation

*T*). In other words, the MOs projected onto their BP AO list exhibit a well-controlled truncation error of 1 –

*T*. (38) For the PD construction, the BP atom lists are obtained for each LMO and PAO with completeness criteria of

*T*

_{PDo}= 0.999 and

*T*

_{PDv}= 0.98, respectively. The SOMOs are part of both the occupied and virtual subspaces; thus, both of these BP atom lists are assembled for them.

*T*

_{PDv}criterion overlaps with the BP list of the LMO, then this SOMO is appended to the spin-down PAO list of the PD. The atom and AO lists of the PD are obtained as the union of the BP lists of all MOs (LMO, PAOs, SOMOs) of the PD. The MOs of the PD are projected onto the AO basis of the PD, and the PAOs (and potential spin-down SOMOs) are orthogonalized within this truncated AO basis, (38) leading to different spin-up and spin-down MOs. Then, for the noniterative evaluation of the MP2 pair energies, the PD’s virtual space is canonicalized separately for the spin-up and spin-down MOs. Finally, the multipole approximated opposite-spin MP2 pair correlation energies (38) are evaluated as

*F*

_{i′i′}(

*F*

_{I′I′}) is the diagonal element of the spin-up (spin-down) Fock matrix. The ERIs of written in the Mulliken notation are obtained using the multipole expansion up to the fourth order, that is, including terms with dipole–dipole, dipole–quadrupole, quadrupole–quadrupole, and dipole–octopole moments. (38)

_{w}is the same strong pair threshold employed in our methods previously, (38,88−90) and

*f*

_{w}is a scaling factor introduced for the following reasons. Let us consider the case when one LMO of the pair, say , is SO. Then, the second and third terms of eq 7 vanish, and therefore, all such pair correlation energies contain only half as many terms compared to the pair energy of two DO LMOs. Furthermore, if both and are SO, then only the first term of eq 7 survives, leading to 4 times less terms contributing to an SO–SO pair than to a DO–DO pair. In accord with this consideration, on average, we find the SO–DO (SO–SO) pair correlation energies to be twice (four times) as small as those of DO–DO pairs. To handle the strong/distant pair characterization of all pair types on an equal footing,

*f*

_{w}factors of 1, (1/2), and (1/4) are employed for the DO–DO, DO–SO, and SO–SO pairs, respectively. The numerical properties of this scaling are analyzed in Section 5.1. We note that a similar scaling factor of (1/3) is introduced in ref (63) in the DLPNO context for pairs involving at least one SOMO. On systems with unusually large numbers of SOMOs, the factor of (1/3) provided better numerical performance than 0.1 or 0.5. (63) This could be explained by the fact that, for the systems explored in ref (63), (1/3) is the closest to the weighted average of (1/2) and (1/4) recommended here.

### 3.5. Extended Domain Construction

*T*

_{EDo}= 0.9999 for all LMOs of the ED. The AOs on these atoms form the AO basis of the ED. The LMOs are projected onto the AOs of their respective BP atom lists ensuring at most 1 –

*T*

_{EDo}truncation error and are then reorthogonalized. The virtual space of the ED is spanned by restricted PAOs originating from atoms of the PAO center domain (PCD) of the ED. The PCD is the union of the more compact BP atom lists of all LMOs in the ED obtained with

*T*

_{o}= 0.985. Since the PAOs tend to be more delocalized than the LMOs, they are projected onto the whole AO basis of the ED. Analogously to the case of the PD construction, the SO LMOs of the ED are appended to the spin-down unoccupied MOs of the ED. The specific combination of the Gram–Schmidt and Löwdin algorithms (112,113) is employed for the orthogonalization of the virtual space of the ED analogously to our previous approach. (38,89) Finally, pseudocanonical and hence unrestricted occupied and virtual orbitals are obtained for the iteration free MP2 energy formulae of the ED.

### 3.6. Integral Transformation in the Extended Domain

**K**ERI tensors are factorized as

*I*

_{ãi′,P}= (

*ãi*′|

*P*) denotes three-center two-electron integrals, and

*P*refers to the auxiliary basis functions. The two-center integral matrix

*V*

_{PQ}= (

*P*|

*Q*) is subjected to Cholesky decomposition (

**V**=

**LL**

^{T}) yielding the

**J**=

**I**(

**L**

^{–1})

^{T}tensor. We showed in ref (38) that the auxiliary basis functions residing on the atoms of the PCD can accurately expand all LMO–PAO orbital product densities of the ED; thus, the auxiliary function list of the ED is chosen accordingly.

**C**and

**P**collect the occupied and virtual MO coefficients discussed in Section 3.5. First, the (

*μ̃ν̃*|

*P*) AO integrals are evaluated for a shell triplet at a time using a highly optimized three-center two-electron AO integral code (116) only for the AOs and auxiliary functions of the ED. These batches are immediately subjected to the first transformation of scheme eq 10, leading to half-transformed integrals with one index in the restricted LMO basis and then discarded. This integral-direct approach effectively makes use of the available memory and data traffic bandwidth between the lower levels of cache and the CPU. The evaluation and first transformation of the three-center ERIs are the most computationally demanding operations in our LMP2 scheme and can be performed at a similar cost as in the closed-shell implementation because restricted LMOs are employed. The introduction of this intermediate step transforming to the restricted LMO basis is thus more effective than transforming from the AO basis directly to the semicanonical occupied basis. The latter, restricted LMO to semicanonical MO transformation is performed much more efficiently as the final step of scheme eq 10. Before that, however, it is beneficial to decrease the number of operations by performing the AO-to PAO transformations (second step of scheme eq 10). Note that the number of integrals entering the second half-transformation is considerably lower than in the first step. Consequently, there is no motivation to perform the AO-to-PAO transformation in two steps by making use of the restricted PAO basis unlike in the case of the first half-transformation. In conclusion, the three-center ERIs are thus transformed to the spin-up and spin-down ED MO bases in a cost comparable to that of the closed-shell alternative.

### 3.7. Energy Contribution in the Extended Domain

*i*′ or

*I*′ index. Thus, we recommended (38,88) circumventing the redundant evaluation of MP1 amplitudes via CD (91) or LT (29,30) techniques. The benefit is that, by factorizing the energy denominators, we can directly evaluate the amplitudes with mixed restricted LMO and semicanonical ED MO indices, e.g.,

_{ĩ}, ε

_{Ĩ}), max(ε

_{ã}, ε

_{Ã})] to determine the weights (

*w*

_{ω}) and quadrature points (

*t*

_{ω}). Then, the

**J̅**integrals of eq 11 can be constructed, e.g., as

*c*

_{ãĩ}

^{ω}denotes the elements of the Cholesky vector, or in the case of LT

*J̅*

_{ãi′,P}

^{ω}obtained from

*J̅*

_{ãĩ,P}

^{ω}via the unitary transformation of the occupied MO index.

*t*

_{i′j̃}

^{ãb̃},

*t*

_{I′J̃}

^{ÃB̃}, and

*t*

_{I′j̃}

^{Ãb̃}) are evaluated analogously using the appropriate spin cases of the

*J̅*tensors. Finally, the RO-LMP2 energy contribution of orbital can be evaluated in its ED as

*t*

_{i′j̃}

^{ãb̃}and

*t*

_{I′J̃}

^{ÃB̃}in the second and fifth terms; thus, the evaluation of eq 14 takes about three times more operations than its closed-shell analogue.

### 3.8. Contribution of Single Excitations

*f*

_{i′}

^{ã}and

*f*

_{I′}

^{Ã}of eq 14. The reason for that is a small contamination of the projected occupied (virtual) orbitals of the ED from the virtual (occupied) subspace spanned by the untruncated MOs of the entire system. In the closed-shell context, we found this source of error small and well-controlled by the BP completeness criteria governing the truncation of the ED’s LMOs. (38) Previously, it was found best not to include these artificial off-diagonal Fock-matrix contributions into the ED’s correlation energy contribution. However, this strategy is more challenging to follow for the open-shell case because one cannot simply discard the correlation energy contributions of the single excitations. To maintain the exact MP2 energy as the approximation-free limit of the present local scheme and to handle the off-diagonal Fock-matrix contributions comparably to the closed-shell case, the two effects are separated as follows.

*f*

_{i′}

^{ã}and

*f*

_{I′}

^{Ã}quantities in each ED only from the off-diagonal part of the original semicanonical Fock matrices. The latter are computed in the AO basis at the end of the complete molecule SCF computation as

**F**and

**F**

^{OD}are the complete Fock matrix and its off-diagonal part in the AO basis, respectively,

**C**holds the unrestricted MO coefficients, and

**ϵ**is a matrix with the corresponding orbital energies on its diagonal. The benefits of storing the additional (spin-up and spin-down)

**F**

^{OD}matrices are illustrated with the example of vitamin E succinate (see Section 4.2). Using the complete

**F**to compute the first and fourth terms of eq 14 would result in a 124% relative error in the singles contribution or in a 0.1% relative error with respect to the total correlation energy. Compared to that, replacing

**F**by

**F**

^{OD}in the calculation of the

*f*

_{i′}

^{ã}and

*f*

_{I′}

^{Ã}matrices, the error in the singles contribution reduces to 0.01%, which is negligible from the perspective of the total correlation energy. For clarity, the complete Fock matrices are employed everywhere else in the algorithm, such as for the semicanonicalization of PD or ED orbitals. The use of

**F**

^{OD}is limited to the energy contribution of single excitations.

### 3.9. Approximate Long-Range Spin Polarization

### 3.10. Scaling of the Algorithm

_{2})

_{n}-Th]

^{2+}diradicals, where Th denotes thiophene rings attached to the end of the alkane chains. (120) Detailed timing data are presented in Section S1 of the Supporting Information. In these measurements, canonical DF-MP2 exhibited an -scaling, which is somewhat lower than its formal -scaling. This can be understood as the most time-consuming step is still the -scaling integral transformation even for the largest chain. In comparison, the LMP2 algorithm exhibits clear linear scaling, which sets in already for the smallest systems. Because of the redundancy-free evaluation of the LMP2 amplitudes, the DF-MP2 and LMP2 calculations take comparable time only up to about 50 atoms followed by the clearly superior performance of LMP2 for larger systems.

## 4. Computational Details and Test Systems

### 4.1. Technicalities

^{–4}.

*E*

_{DF-MP2}

^{c}) are obtained as (100%)·(

*E*

_{LMP2}

^{c}–

*E*

_{DF-MP2}

^{c})/

*E*

_{DF-MP2}

^{c}.

### 4.2. Benchmark Sets and Test Systems

_{12}(5′-deoxyadenosylcobalamin, dAdoCbl) with open-shell systems of up to 179 atoms [the Cob

^{II}alamin (Cbl) radical] was also considered (132) (see Figure 3).

_{2}. The diradical reactant state of Figure 5 results from a single electron transfer from reduced FAD to O

_{2}, leading finally to the oxidized form of FAD and H

_{2}O

_{2}. Models of the corresponding triplet and singlet states of the structures labeled by O1

^{T}and O3

^{CSS}in ref (133) are provided in the Supporting Information.

## 5. Accuracy of the Local Approximations

_{w}and

*T*

_{EDo}) responsible for the bulk of the local error. Open-shell-specific approximations, which did not appear before, are also thoroughly benchmarked. For the remaining truncation parameters, which affect the closed- and open-shell systems similarly, such as the BP parameters of the PDs or the order of multipole expansion, the previously assessed values are adopted. (38,89) Note that such approximations are also active and hence tested in the benchmarks of Section 6.

### 5.1. Strong Pair Classification

_{w}) by

*f*

_{w}factors of (1/2) and (1/4) for the pairs including one or two SOMO(s), respectively.

*I*and

*J*. The values are collected from multiple systems containing two methyl carbene species placed at varying distances from each other, with both methyl carbene subsystems being in their local triplet state. The left panel, collecting unscaled pair energies, illustrates that pairs involving different numbers of SOMOs gather into three distinct clusters of points. This verifies our expectation that for pairs with comparable orbital center distances, smaller pair correlation energies are obtained for SO–SO or DO–SO pairs than for DO–DO pairs. Consequently, the curves of the three groups of unscaled pair energies intersect the default pair energy threshold (dashed horizontal line) at different distances. This reveals a potential bias in the strong/distant pair classification of pairs involving SOMOs. However, our goal is to ensure comparable classification for all pairs exhibiting a similar pair distance or interaction strength regardless of their occupation. To that end, we examine the distance dependence of the same pair correlation energies scaled by , that is, by 2 and 4 for the DO–SO and SO–SO pairs, respectively. This emulates the use of the

*f*

_{w}ε

_{w}strong pair threshold instead of ε

_{w}. The resulting scaled pair energies collected in the right panel of Figure 6 indeed exhibit the same trend for all three types of pairs independent of the occupation. Another beneficial consequence of using the scaled pair threshold is that the chance of including the SOMOs in the EDs increases. These SOMOs often play an important role in the chemical processes of open-shell species, and therefore, their improved description is advantageous.

### 5.2. Strong Pair Selection

_{w}). To that end, LMP2 calculations are performed in which all local approximations are turned off except for the strong pair criterion of the ED construction. The approximations governed by this threshold are negligible for small systems and start to operate to a considerable extent for larger molecules. Besides the correlation energies of such extended systems (42–81 atoms), the accuracy of three different kinds of relative energies is also assessed: the vertical ionization potential (VIP) of testosterone, the radical stabilization reaction energy (RSE) of vitamin E succinate, and the singlet–triplet (S–T) gap for artemisinin. The basis set of aug-cc-pVTZ is used for all species so that the tests will be performed with a large basis set including diffuse functions sufficient for realistic applications. Diffuse AOs are more challenging to handle for local approximations, and consequently, such AOs cannot be omitted in representative convergence tests.

_{w}. Rapid convergence is observed for all cases, similar to previous findings on closed-shell systems. (38,89) The energy differences are practically converged already at the default ε

_{w}= 10

^{–5}

*E*

_{h}setting, and the largest error of 0.05 kcal/mol is negligible compared to the 217 kcal/mol VIP of testosterone. The corresponding correlation energies are also accurate up to 0.03% relative errors with this default threshold.

_{w}= 10

^{–5}

*E*

_{h}corresponds to the tighter settings employed in ref (38), and it has been employed also as default in the context of our LNO-CC approaches (89,90) and also with the LMP2 scheme since 2018. The strong pair selection and ED construction controlled by ε

_{w}= 10

^{–5}

*E*

_{h}were found to be similarly accurate previously for a number of alternative systems containing up to 260 atoms and for various reaction and interaction energies involving closed-shell systems. (38,89,90,134,135)

### 5.3. Representation of the LMOs

*T*

_{EDo}). Together with ε

_{w}, these two thresholds also determine the number of atoms, AOs, and the truncation errors of the MOs in the ED.

*T*

_{EDo}parameter are performed for the same open-shell species and energy differences as used in Section 5.2 for ε

_{w}. Again, only the local approximation corresponding to

*T*

_{EDo}was active, and all other approximations were turned off to separate the effect of

*T*

_{EDo}.

*T*

_{EDo}toward the DF-MP2 reference. Both the correlation energies and the energy differences are converged already at

*T*

_{EDo}= 0.9999 (1 –

*T*

_{EDo}= 10

^{–4}in Figure 8), which is chosen as default. We note again that this value corresponds to the tighter setting introduced in ref (38), and it is chosen as default also in our recent closed-shell LMP2 as well as LNO-CC methods. (89,90)

### 5.4. Assessment of the Long-Range Spin-Polarization Approximation

error in energy difference | |||||||
---|---|---|---|---|---|---|---|

atoms | LMOs | E_{LMP2}^{c} error [%] | [cal/mol] | [%] | EDs without SOMOs [%] | ||

vitamin E succinate | 81 | 89 | 7.8 × 10^{–7} | 0.027 | 2.8 × 10^{–4} | 54 | |

FeC_{72}N_{2}H_{100} | ^{5}A | 175 | 205 | 1.7 × 10^{–5} | 0.66 | 1.4 × 10^{–3} | 54 |

^{3}A | 204 | 8.5 × 10^{–6} | 54 | ||||

Cbl radical | 179 | 250 | 7.7 × 10^{–6} | 0.81 | 1.6 × 10^{–3} | 68 | |

bicarbonate | ^{5}A | 565 | 789 | 1.2 × 10^{–4} | 3.5 | 8.7 × 10^{–3} | 91 |

^{3}A | 788 | 1.1 × 10^{–4} | 92 | ||||

DAAO | 601 | 838 | 2.3 × 10^{–7} | 0.078 | 2.6 × 10^{–4} | 76 |

^{a}

See the text for explanation.

^{–4}–10

^{–7}% for all cases are surprisingly small. This error range is comparable to or even better than that of any other employed approximation, including the DF approach. Consequently, most of the energy differences are also practically unaffected by this approximation being below 1 cal/mol for all but one example.

*Ŝ*

^{2}eigenfunction as the reference, the QRO reference energy, and potentially also the corresponding unrestricted Fock-matrix elements, may differ from the completely variationally optimized UHF solution more than the analogous ROHF-based quantities. Therefore, the approximation of QRO-based unrestricted Fock-matrix elements by spin-averaged ones may affect the interaction of the bicarbonate’s SOMOs with the rest of the DOMOs in a somewhat more pronounced manner.

## 6. Benchmarks for Small and Medium-Sized Systems

### 6.1. Accuracy of Correlation Energies

basis | error measure | error in E_{LMP2}^{c} [%] | error in RSE [kcal/mol] |
---|---|---|---|

aug-cc-pV(T + d)Z | MAX | 0.014 | 0.041 |

MAE | 0.003 | 0.010 | |

STD | 0.003 | 0.011 | |

aug-cc-pV(Q + d)Z | MAX | 0.016 | 0.065 |

MAE | 0.003 | 0.029 | |

STD | 0.004 | 0.012 | |

CBS(T,Q) | MAX | 0.017 | 0.097 |

MAE | 0.004 | 0.055 | |

STD | 0.005 | 0.020 |

basis | error measure | error in E_{LMP2}^{c} [%] | error in IP [meV] |
---|---|---|---|

aug-cc-pV(T + d)Z | MAX | 0.016 | 2.03 |

MAE | 0.004 | 0.47 | |

STD | 0.005 | 0.60 |

basis | error measure | error in E_{LMP2}^{c} [%] | error in S–T gap [kcal/mol] |
---|---|---|---|

cc-pVDZ | MAX | 0.04 | 0.13 |

MAE | 0.03 | 0.06 | |

STD | 0.01 | 0.04 | |

cc-pVTZ | MAX | 0.04 | 0.13 |

MAE | 0.02 | 0.05 | |

STD | 0.01 | 0.04 |

_{72}N

_{2}H

_{100}complex, but again this consistency leads to a negligible error in the spin-state splitting. Considering that the average system size increases by about 10 times when stepping from smaller to larger systems, the size dependence of the relative accuracy also appears excellent well above the size range where all approximations start to operate to their full extent.

molecule | atoms | no. of AOs | E_{LMP2}^{c} error [%] | ΔE error [kcal/mol] | time^{b} [min] | |
---|---|---|---|---|---|---|

glutathione ion | 37 | 1320 | 0.05 | –0.05 | 21 | |

artemisinin ^{3}A | 42 | 1426 | 0.04 | –0.05 | 70 | |

testosterone ion | 49 | 1610 | 0.05 | –0.01 | 77 | |

borrelidin ion | 78 | 2599 | 0.08 | 0.12 | 256 | |

vitamin E succinate | 81 | 2553 | 0.05 | 0.01 | 89 | |

[Th–(CH_{2})_{50}–Th]^{2+} | 166 | 2508^{c} | 0.05 | 5 | ||

FeC_{72}N_{2}H_{100} | ^{5}A | 175 | 2939^{c} | 0.11 | 0.005 | 180 |

^{3}A | 0.11 | 186 |

^{a}

Unless otherwise noted, the calculations were carried out with the aug-cc-pV(T + d)Z basis set.

^{b}

Wall-clock times measured on an 8-core 3 GHz Intel Xeon E5-1660 processor.

^{c}

The def2-TZVP basis set was utilized.

### 6.2. Radical Stabilization Energies

^{•}denotes various radicals containing C, N, O, F, P, and S atoms. The MAEs of the LMP2 RSEs collected in Table 3 are below 0.03 kcal/mol for the aug-cc-pV(X + d)Z basis set with both X = T and X = Q, while the CBS extrapolation slightly increases the MAE to 0.05 kcal/mol. The corresponding MAX errors of 0.04, 0.06, and 0.10 kcal/mol at the triple-ζ, quadruple-ζ, and CBS(T,Q) levels, respectively, are still well within the intrinsic accuracy of MP2. The STD values of 0.01–0.02 kcal/mol underline the reproducibility of the excellent accuracy. One can also compare the accuracy of the present LMP2 results to those obtained with PNO-ROMP2 in ref (65) for the same structures and with the same aug-cc-pV(T + d)Z basis set. The two approaches perform similarly well; in terms of the MAX and MAE measures compared to the respective references, LMP2 is somewhat more accurate than PNO-ROMP2 and slightly worse than the explicitly correlated PNO-ROMP2 variant.

### 6.3. Ionization Potentials

### 6.4. Singlet–Triplet Energy Gaps

### 6.5. Energy Differences for Larger Systems

_{72}N

_{2}H

_{100}complex. It is reassuring that none of the RSE or spin-state gap errors exceed the corresponding MAEs obtained for the same properties but with much smaller systems. Regarding the IPs, only the still highly acceptable 0.12 kcal/mol error of borrelidin exceeds the inaccuracies obtained for the IP test compilation. Thus, as expected from the underlying accurate LMP2 correlation energies, we do not find any increase in the inaccuracy of the inspected energy differences in spite of the considerable growth in system size.

_{72}N

_{2}H

_{100}complex, the remaining energy differences involve both open- and closed-shell species. Consequently, the performance of LMP2 is balanced irrespective of the presence of SOMOs, allowing for the investigation of chemical processes involving both open- and closed-shell species.

## 7. Representative Applications and Computational Requirements

_{72}N

_{2}H

_{100}complex and the Cbl radical of 175–179 atoms and of about 3000 AOs constitute the first group, as these systems are close to the capability limits of efficient open-shell DF-MP2 implementations. An additional similarity is that both systems contain a transition-metal atom, and the corresponding SOMO(s) are located close to the center of the molecule, resulting in a large number of strong pairs involving SOMO(s). The 21–25% strong pair ratio is indeed noticeably higher than the 16% obtained for the closed-shell vancomycin molecule of the same size (176 atoms) with the same settings. (38) The corresponding EDs containing on an average (at most) about 115 (167) atoms are also significantly larger than the EDs of vancomycin built with 72 (129) atoms.

molecule | FeC_{72}N_{2}H_{100} | Cbl radical | bicarbonate | DAAO | ||
---|---|---|---|---|---|---|

atoms | 175 | 179 | 565 | 601 | ||

LMOs | 205 | 250 | 788 | 837 | 838 | |

SOMOs | 4 | 1 | 4 | 0 | 2 | |

AO basis | def2-TZVP | def2-TZVP | def2-SVP’ | def2-TZVP | def2-TZVP | |

basis functions | 2939 | 3369 | 5434 | 10 560 | 11 006 | |

auxiliary functions | 7306 | 8379 | 17 782 | 26 064 | 27 071 | |

strong pairs [%] | 25 | 21 | 6.3 | 6.8 | 5.9 | 5.9 |

[%] | 0.19 | 0.19 | 0.28 | 0.25 | 0.25 | 0.25 |

atoms in ED | 114 (165) | 116 (169) | 138 (317) | 132 (295) | 124 (268) | 137 (353) |

AOs in ED | 2086 (2854) | 2342 (3278) | 1376 (3195) | 2634 (6015) | 2449 (5408) | 2693 (6943) |

PAOs in ED | 1020 (1812) | 1017 (1828) | 481 (1052) | 973 (2037) | 864 (1884) | 902 (2930) |

type of reference | ROHF | ROHF | QRO (UHF) | RHF | ROHF | |

DF-HF energy | –4156.159945 | –5878.796625 | –15182.8673^{c} | –15197.9344^{c} | –14740.9398 | –14740.9040 |

LMP2 energy | –12.3329 | –16.4723 | –43.2442 | –52.3847 | –55.3431 | –55.3319 |

HF (1 iteration) | 28 | 43 | 29^{b} | 183^{b} | 152^{b} | 157^{b} |

localization | 0.1 | 0.3 | 4.8 | 4.3 | 2.8 | 3.4 |

pair energies | 1.2 | 8.7 | 38 | 4.8 | 11 | 48 |

integral trf. | 56 | 157 | 188 | 451 | 374 | 639 |

amplitudes & E_{LMP2}^{c} | 38 | 98 | 18 | 100 | 54 | 213 |

total LMP2 | 95 | 264 | 245 | 557 | 439 | 900 |

memory req. | 9.8 | 10 | 4.6 | 17 | 6.7 | 45 |

^{a}

Using a 20-core 1.3 GHz Intel Xeon Gold 6138 CPU.

^{b}

Using the default local fitting domain size. The final iteration with larger fitting domains took about 3.5–4.8 times longer.

^{c}

DF-HF energies calculated with semicanonical QRO orbitals.

_{72}N

_{2}H

_{100}complex with both RO-LMP2 and the corresponding DF-MP2 reference is in good agreement with the 2.018, 1.852, and 2.120 eV values reported with PNO-RMP2, (65) NEVPT2, (120) and CASPT2, (136) respectively. It is interesting to realize that the LMP2/def2-TZVP value of 1.759 eV obtained for the quintet–triplet gap of bicarbonate is considerably lower because of the markedly different ligand field of its Fe(II) center. While the slow basis set convergence issue of electron correlation calculations is well-known, the insufficient level of AO basis completeness provided by double-ζ-quality basis sets should still be pointed out as frequently as possible.

def2-SVP | def2-TZVP | |||||
---|---|---|---|---|---|---|

HF | ΔE_{LMP2}^{c} | ΔE_{LMP2}^{total} | HF | ΔE_{LMP2}^{c} | ΔE_{LMP2}^{total} | |

FeC_{72}N_{2}H_{100}^{5}A–^{3}A gap | 57.52 | –8.92 | 48.60 | 57.56 | –10.73 | 46.82 |

bicarbonate ^{5}A–^{3}A gap | 52.62 | –12.35 | 40.26 | 52.67 | –12.11 | 40.56 |

Cbl + Ado → dAdoCbl | –43.38 | 99.84 | 56.46 | –50.41 | 102.13 | 51.73 |

DAAO | 20.46 | 8.71 | 29.17 | 22.44 | 7.08 | 29.52 |

## 8. Summary and Conclusions

_{2}via d-amino acid oxidase is also challenging because of a poorly localized SOMO spreading over an entire flavin moiety. We anticipate that the common target applications of RO-LMP2 will be significantly simpler. However, it is satisfactory that such complicated systems have also become routinely available, especially if the single-node (20-core) RO-LMP2 runtimes of 9–15 h are considered. Consequently, the presented local approximations extend the reach of open-shell MP2 as well as of spin-scaled MP2 and DH DFT methods to systems of 500–600 atoms with reasonable basis sets. Except for potential bottlenecks in the ROHF/ROKS optimization, RO-LMP2 should also be applicable for even larger molecules approaching the limit of our closed-shell LMP2 and LNO-CCSD(T) codes, which is currently about 1000–2000 atoms and 45 000 atomic orbitals. (89,90)

## Supporting Information

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.1c00093.

Linear-scaling time measurements, list of default settings for the local approximations, PES of the ethane-1,2-diphenyl radical, reference energies for the RSE30, IP21, and AC12 test sets and the larger molecules considered as well as the accuracy assessment of the SCS-LMP2 and LB2PLYP methods; structures of the IP21 ions and DAAO models are also provided (PDF)

## Terms & Conditions

Most electronic Supporting Information files are available without a subscription to ACS Web Editions. Such files may be downloaded by article for research use (if there is a public use license linked to the relevant article, that license may permit other uses). Permission may be obtained from ACS for other uses through requests via the RightsLink permission system: http://pubs.acs.org/page/copyright/permissions.html.

## Acknowledgments

Helpful discussions with Qianli Ma regarding the structures used for the RSE test set, with Masaaki Saitow and Ashutosh Kumar regarding the bicarbonate SCF computations of refs (63) and (98), and with Dóra J. Kiss regarding the DAAO structures are gratefully acknowledged. The authors are grateful for the financial support from the National Research, Development, and Innovation Office (NKFIH, Grant No. KKP126451). The research reported in this paper and carried out at BME has been supported by the NRDI Fund (TKP2020 IES, Grant No. BME-IE-BIO) based on the charter of bolster issued by the NRDI Office under the auspices of the Ministry for Innovation and Technology. The work of PRN is supported by the ÚNKP-19-4 and ÚNKP-20-5 New National Excellence Program of the Ministry for Innovation and Technology from the source of the National Research, Development and Innovation Fund and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. The computing time granted on the Hungarian HPC Infrastructure at NIIF Institute, Hungary, and the DECI resource Saga based in Norway at Trondheim with support from the PRACE aisbl (NN9914K) are gratefully acknowledged.

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(American Institute of Physics)A simple modification of the second-order Moller-Plesset perturbation theory (MP2) to improve the description of mol. ground state energies is proposed. The total MP2 correlation energy is partitioned into parallel- and antiparallel-spin components which are sep. scaled. The two parameters (scaling factors), whose values can be justified by basic theor. arguments, were optimized on a benchmark set of 51 reaction energies composed of 74 first-row mols. The new method performs significantly better than std. MP2: the rms [mean abs. error (MAE)] deviation drops from 4.6 (3.3) to 2.3 (1.8) kcal/mol. The max. error is reduced from 13.3 to 5.1 kcal/mol. Significant improvements are esp. obsd. for cases which are usually known as MP2 pitfalls while cases already described well with MP2 remain almost unchanged. Even for 11 atomization energies not considered in the fit, uniform improvements [MAE: 8.1 kcal/mol (MP2) vs. 3.2 kcal/mol (new)] were found. 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The parameters are obtained in a least-squares-fit procedure to the G2/97 set of heat of formations. Opposed to conventional hybrid functionals, the optimum ax is found to be quite large (53% with c = 27%) which at least in part explains the success for many problematic mol. systems compared to conventional approaches. The performance of the new functional termed B2-PLYP is assessed by the G2/97 std. benchmark set, a second test suite of atoms, mols., and reactions that are considered as electronically very difficult (including transition-metal compds., weakly bonded complexes, and reaction barriers) and comparisons with other hybrid functionals of GGA and meta-GGA types. According to many realistic tests, B2-PLYP can be regarded as the best general purpose d. functional for mols. (e.g., a mean abs. deviation for the two test sets of only 1.8 and 3.2 kcal/mol compared to about 3 and 5 kcal/mol, resp., for the best other d. functionals). Very importantly, also the max. and minium errors (outliers) are strongly reduced (by about 10-20 kcal/mol). Furthermore, very good results are obtained for transition state barriers but unlike previous attempts at such a good description, this definitely comes not at the expense of equil. properties. Preliminary calcns. of the equil. bond lengths and harmonic vibrational frequencies for diat. mols. and transition-metal complexes also show very promising results. The uniformity with which B2-PLYP improves for a wide range of chem. systems emphasizes the need of (virtual) orbital-dependent terms that describe nonlocal electron correlation in accurate exchange-correlation functionals. From a practical point of view, the new functional seems to be very robust and it is thus suggested as an efficient quantum chem. method of general purpose.**17**Sancho-García, J. C.; Adamo, C. Double-hybrid density functionals: Merging wavefunction and density approaches to get the best of both worlds.*Phys. Chem. Chem. Phys.*2013,*15*, 14581, DOI: 10.1039/c3cp50907aGoogle Scholar17https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXht1KhtbvF&md5=4216f41fe053cdc2348840cbd2567f0cDouble-hybrid density functionals: merging wavefunction and density approaches to get the best of both worldsSancho-Garcia, J. C.; Adamo, C.Physical Chemistry Chemical Physics (2013), 15 (35), 14581-14594CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)A review. We review why and how double-hybrid d. functionals have become new leading actors in the field of computational chem., thanks to the combination of an unprecedented accuracy together with large robustness and reliability. Similar to their predecessors, the widely employed hybrid d. functionals, they are rooted in the Adiabatic Connection Method from which they emerge in a natural way. We present recent achievements concerning applications to chem. systems of the most interest, and current extensions to deal with challenging issues such as non-covalent interactions and excitation energies. These promising methods, despite a slightly higher computational cost than other typical d.-based models, are called to play a key role in the near future and can thus pave the way towards new discoveries or advances.**18**Goerigk, L.; Grimme, S. Double-hybrid density functionals.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2014,*4*, 576, DOI: 10.1002/wcms.1193Google Scholar18https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhvVelu7jI&md5=c2c8a4d2d17cea5bc4a9c559d42742c8Double-hybrid density functionalsGoerigk, Lars; Grimme, StefanWiley Interdisciplinary Reviews: Computational Molecular Science (2014), 4 (6), 576-600CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)Double-hybrid d. functionals (DHDFs) are reviewed in this study. In DHDFs parts of conventional d. functional theory (DFT) exchange and correlation are replaced by contributions from nonlocal Fock-exchange and second-order perturbative correlation. The latter portion is based on the well-known MP2 wave-function approach in which, however, Kohn-Sham orbitals are used to calc. its contribution. First, related methods preceding this idea are reviewed, followed by a thorough discussion of the first modern double-hybrid B2-PLYP. Parallels and differences between B2-PLYP and its various successors are then outlined. This discussion is rounded off with representative thermochem. examples demonstrating that DHDFs belong to the most robust and accurate DFT approaches currently available. This anal. also presents hitherto unpublished results for recently developed DHDFs. Finally, how double-hybrids can be combined with linear-response time-dependent DFT is also outlined and the value of this approach for electronically excited states is shown. WIREs Comput Mol Sci 2014, 4:576-600. doi: 10.1002/wcms.1193 For further resources related to this article, please visit the . Conflict of interest: The authors have declared no conflicts of interest for this article.**19**Martin, J. M. L.; Santra, G. Empirical Double-Hybrid Density Functional Theory: A ‘Third Way’ in Between WFT and DFT.*Isr. J. Chem.*2020,*60*, 787, DOI: 10.1002/ijch.201900114Google Scholar19https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXitlyhsrnM&md5=182610d9f5560d261abf36f80a6d9d2eEmpirical Double-Hybrid Density Functional Theory: A 'Third Way' in Between WFT and DFTMartin, Jan M. L.; Santra, GolokeshIsrael Journal of Chemistry (2020), 60 (8-9), 787-804CODEN: ISJCAT; ISSN:0021-2148. (Wiley-VCH Verlag GmbH & Co. KGaA)A review. Double hybrid d. functional theory arguably sits on the seamline between wavefunction methods and DFT: it represents a special case of Rung 5 on the "Jacob's Ladder" of John P. Perdew. For large and chem. diverse benchmarks such as GMTKN55, empirical double hybrid functionals with dispersion corrections can achieve accuracies approaching wavefunction methods at a cost not greatly dissimilar to hybrid DFT approaches, provided RI-MP2 and/or another MP2 acceleration techniques are available in the electronic structure code. Only a half-dozen or fewer empirical parameters are required. For vibrational frequencies, accuracies intermediate between CCSD and CCSD(T) can be achieved, and performance for other properties is encouraging as well. Organometallic reactions can likewise be treated well, provided static correlation is not too strong. Further prospects are discussed, including range-sepd. and RPA-based approaches.**20**Chai, J.-D.; Head-Gordon, M. Long-range corrected double-hybrid density functionals.*J. Chem. Phys.*2009,*131*, 174105 DOI: 10.1063/1.3244209Google Scholar20https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhtl2hsLrO&md5=4a5e3b76f67c32897d242da653f7a47cLong-range corrected double-hybrid density functionalsChai, Jeng-Da; Head-Gordon, MartinJournal of Chemical Physics (2009), 131 (17), 174105/1-174105/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We extend the range of applicability of our previous long-range cor. (LC) hybrid functional, ωB97X, with a nonlocal description of electron correlation, inspired by second-order Moller-Plesset (many-body) perturbation theory. This LC "double-hybrid" d. functional, denoted as ωB97X-2, is fully optimized both at the complete basis set limit (using 2-point extrapolation from calcns. using triple and quadruple zeta basis sets), and also sep. using the somewhat less expensive 6-311++G(3df,3pd) basis. On independent test calcns. (as well as training set results), ωB97X-2 yields high accuracy for thermochem., kinetics, and noncovalent interactions. In addn., owing to its high fraction of exact Hartree-Fock exchange, ωB97X-2 shows significant improvement for the systems where self-interaction errors are severe, such as sym. homonuclear radical cations. (c) 2009 American Institute of Physics.**21**Goerigk, L.; Grimme, S. Efficient and Accurate Double-Hybrid-Meta-GGA Density Functionals—Evaluation with the Extended GMTKN30 Database for General Main Group Thermochemistry, Kinetics, and Noncovalent Interactions.*J. Chem. Theory Comput.*2011,*7*, 291, DOI: 10.1021/ct100466kGoogle Scholar21https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXhs1Srsb7N&md5=bd9fde6f59698f9f9f7a41195e6ad144Efficient and Accurate Double-Hybrid-Meta-GGA Density Functionals-Evaluation with the Extended GMTKN30 Database for General Main Group Thermochemistry, Kinetics, and Noncovalent InteractionsGoerigk, Lars; Grimme, StefanJournal of Chemical Theory and Computation (2011), 7 (2), 291-309CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present an extended and improved version of our recently published database for general main group thermochem., kinetics, and noncovalent interactions, which is dubbed GMTKN30. Furthermore, we suggest and investigate two new double-hybrid-meta-GGA d. functionals called PTPSS-D3 and PWPB95-D3. PTPSS-D3 is based on reparameterized TPSS exchange and correlation contributions; PWPB95-D3 contains reparameterized PW exchange and B95 parts. Both functionals contain fixed amts. of 50% Fock-exchange. Furthermore, they include a spin-opposite scaled perturbative contribution and are combined with our latest atom-pairwise London-dispersion correction. When evaluated with the help of the Laplace transformation algorithm, both methods scale as N4 with system size. The functionals are compared with the double hybrids B2PLYP-D3, B2GPPLYP-D3, DSD-BLYP-D3, and XYG3 for GMTKN30 with a quadruple-ζ basis set. PWPB95-D3 and DSD-BLYP-D3 are the best functionals in our study and turned out to be more robust than B2PLYP-D3 and XYG3. Furthermore, PWPB95-D3 is the least basis set dependent and the best functional at the triple-ζ level. For the example of transition metal carbonyls, it is shown that, mainly due to the lower amt. of Fock-exchange, PWPB95-D3 and PTPSS-D3 are better applicable than the other double hybrids. Finally, we discuss in some detail the XYG3 functional, which makes use of B3LYP orbitals and electron densities. We show that it is basically a highly nonlocal variant of B2PLYP and that its partially good performance is mainly due to a larger effective amt. of perturbative correlation compared to other double hybrids. We finally recommend the PWPB95-D3 functional in general chem. applications.**22**Zhang, I. Y.; Xu, X.; Jung, Y.; Goddard, W. A. A fast doubly hybrid density functional method close to chemical accuracy using a local opposite spin ansatz.*Proc. Natl. Acad. Sci. U.S.A.*2011,*108*, 19896, DOI: 10.1073/pnas.1115123108Google Scholar22https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhs12gtbvL&md5=0416b2a69fd6724216a249add03a6385A fast doubly hybrid density functional method close to chemical accuracy using a local opposite spin ansatzZhang, Igor Ying; Xu, Xin; Jung, Yousung; Goddard, William A., IIIProceedings of the National Academy of Sciences of the United States of America (2011), 108 (50), 19896-19900, S19896/1-S19896/10CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)We develop and validate the XYGJ-OS functional, based on the adiabatic connection formalism and Girling-Levy perturbation theory to second order and using the opposite-spin (OS) ansatz combined with locality of electron correlation. XYGJ-OS with local implementation scales as N3 with an overall accuracy of 1.28 kcal/mol for thermochem., bond dissocn. energies, reaction barrier heights, and nonbonded interactions, comparable to that of 1.06 kcal/mol for the accurate coupled-cluster based G3 method (scales as N7) and much better than many popular d. functional theory methods: B3LYP (4.98), PBEO (4.36), and PBE (12.10).**23**Zhang, I. Y.; Su, N. Q.; Brémond, É. A. G.; Adamo, C.; Xu, X. Doubly hybrid density functional xDH-PBE0 from a parameter-free global hybrid model PBE0.*J. Chem. Phys.*2012,*136*, 174103 DOI: 10.1063/1.3703893Google Scholar23https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38Xmt12nurw%253D&md5=2f4ea470e5efa8d7da9033f90d27535aDoubly hybrid density functional xDH-PBE0 from a parameter-free global hybrid model PBE0Zhang, Igor Ying; Su, Neil Qiang; Bremond, Eric A. G.; Adamo, Carlo; Xu, XinJournal of Chemical Physics (2012), 136 (17), 174103/1-174103/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Following the XYG3 model which uses orbitals and d. from B3LYP, an empirical doubly hybrid (DH) functional is developed by using inputs from PBE0. This new functional, named xDH-PBE0, has been tested on a no. of different mol. properties, including atomization energies, bond dissocn. enthalpies, reaction barrier heights, and nonbonded interactions. From the results obtained, xDH-PBE0 not only displays a significant improvement with respect to the parent PBE0, but also shows a performance that is comparable to XYG3. Arguably, while PBE0 is a parameter-free global hybrid (GH) functional, the B3LYP GH functional contains eight fit parameters. From a more general point of view, the present work points out that reliable and general-purpose DHs can be obtained with a limited no. of fit parameters. (c) 2012 American Institute of Physics.**24**Kozuch, S.; Martin, J. M. L. Spin-Component-Scaled Double Hybrids: An Extensive Search for the Best Fifth-Rung Functionals Blending DFT and Perturbation Theory.*J. Comput. Chem.*2013,*34*, 2327, DOI: 10.1002/jcc.23391Google Scholar24https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXhtlans7fN&md5=9d34bb3f6cd3fda3be5a41d58ef93075Spin-component-scaled double hybrids: An extensive search for the best fifth-rung functionals blending DFT and perturbation theoryKozuch, Sebastian; Martin, Jan M. L.Journal of Computational Chemistry (2013), 34 (27), 2327-2344CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)Following up on an earlier preliminary communication (Kozuch and Martin, Phys. Chem. Chem. Phys. 2011, 13, 20104), we report here in detail on an extensive search for the most accurate spin-component-scaled double hybrid functionals [of which conventional double hybrids (DHs) are a special case]. Such fifth-rung functionals approach the performance of composite ab initio methods such as G3 theory at a fraction of their computational cost, and with anal. derivs. available. In this article, we provide a crit. anal. of the variables and components that maximize the accuracy of DHs. These include the selection of the exchange and correlation functionals, the coeffs. of each component [d. functional theory (DFT), exact exchange, and perturbative correlation in both the same spin and opposite spin terms], and the addn. of an ad-hoc dispersion correction; we have termed these parametrizations "DSD-DFT" (Dispersion cor., Spin-component scaled, Double-hybrid DFT). Somewhat surprisingly, the quality of DSD-DFT is only mildly dependent on the underlying DFT exchange and correlation components, with even DSD-LDA yielding respectable performance. Simple, nonempirical GGAs appear to work best, whereas meta-GGAs offer no advantage (with the notable exception of B95c). The best correlation components appear to be, in that order, B95c, P86, and PBEc, while essentially any good GGA exchange yields nearly identical results. On further validation with a wider variety of thermochem., weak interaction, kinetic, and spectroscopic benchmarks, we find that the best functionals are, roughly in that order, DSD-PBEhB95, DSD-PBEP86, DSD-PBEPW91, and DSD-PBEPBE. In addn., DSD-PBEP86 and DSD-PBEPBE can be used without source code modifications in a wider variety of electronic structure codes. Sample job decks for several commonly used such codes are supplied as electronic Supporting Information. Copyright © 2013 Wiley Periodicals, Inc.**25**Brémond, É.; Ciofini, I.; Sancho-García, J. C.; Adamo, C. Nonempirical Double-Hybrid Functionals: An Effective Tool for Chemists.*Acc. Chem. Res.*2016,*49*, 1503, DOI: 10.1021/acs.accounts.6b00232Google Scholar25https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xht1ynsb7O&md5=f2722f267174ead8e7ab3e5daf92a76bNonempirical Double-Hybrid Functionals: An Effective Tool for ChemistsBremond, Eric; Ciofini, Ilaria; Sancho-Garcia, Juan Carlos; Adamo, CarloAccounts of Chemical Research (2016), 49 (8), 1503-1513CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)A review. D. functional theory (DFT) emerged in the last two decades as the most reliable tool for the description and prediction of properties of mol. systems and extended materials, coupling in an unprecedented way high accuracy and reasonable computational cost. This success rests also on the development of more and more performing d. functional approxns. (DFAs). Indeed, the Achilles' heel of DFT is represented by the exchange-correlation contribution to the total energy, which, being unknown, must be approximated. Since the beginning of the 1990s, global hybrids (GH) functionals, where an explicit dependence of the exchange-correlation energy on occupied Kohn-Sham orbitals is introduced thanks to a fraction of Hartree-Fock-like exchange, imposed themselves as the most reliable DFAs for chem. applications. However, if these functionals normally provide results of sufficient accuracy for most of the cases analyzed, some properties, such as thermochem. or dispersive interactions, can still be significantly improved. A possible way out is represented by the inclusion, into the exchange-correlation functional, of an explicit dependence on virtual Kohn-Sham orbitals via perturbation theory. This leads to a new class of functionals, called double-hybrids (DHs). In this Account, we describe our nonempirical approach to DHs, which, following the line traced by the Perdew-Burke-Ernzerhof approach, allows for the definition of a GH (PBE0) and a DH (QIDH) model. In such a way, a whole family of nonempirical functionals, spanning on the highest rungs of the Perdew's quality scale, is now available and competitive with other-more empirical-DFAs. Discussion of selected cases, ranging from thermochem. and reactions to weak interactions and excitation energies, not only show the large range of applicability of nonempirical DFAs, but also underline how increasing the no. of theor. constraints parallels with an improvement of the DFA's numerical performances. This fact further consolidates the strong theor. framework of nonempirical DFAs.Finally, even if nonempirical DH approaches are still computationally expensive, relying on the fact that they can benefit of all tech. enhancements developed for speeding up post-Hartree-Fock methods, there is substantial hope for their near future routine application to the description and prediction of complex chem. systems and reactions.**26**Su, N. Q.; Xu, X. The XYG3 Type of Doubly Hybrid Density Functionals.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2016,*6*, 721, DOI: 10.1002/wcms.1274Google Scholar26https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhslKgsbfP&md5=be9356f372be073ff2a1794c47a9fc11The XYG3 type of doubly hybrid density functionalsSu, Neil Qiang; Xu, XinWiley Interdisciplinary Reviews: Computational Molecular Science (2016), 6 (6), 721-747CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)Doubly hybrid (DH) functionals have emerged as a new class of d. functional approxns. (DFAs), which not only have a nonlocal orbital-dependent component in the exchange part, but also incorporate the information of unoccupied orbitals in the correlation part, being at the top rung of Perdew's view of Jacob's ladder in DFAs. This review article focuses on the XYG3 type of DH (xDH) functionals, which use a low rung functional to perform the self-consistent-field calcn. to generate orbitals and densities, with which a top rung DH functional is used for final energy evaluation. We will discuss the theor. background of the xDH functionals, briefly reviewing the adiabatic connection formalism, coordinate scaling relations, and Goerling-Levy perturbation theory. General performance of the xDH functionals will be presented for both energies and structures. In particular, we will present the fractional charge behaviors of the xDH functionals, examg. the self-interaction errors, the delocalization errors and the deviation from the linearity condition, as well as their effects on the predicted ionization potentials, electron affinities and fundamental gaps. This provides a theor. rationale for the obsd. good performance of the xDH functionals. WIREs Comput Mol Sci 2016, 6:721-747. doi: 10.1002/wcms.1274 For further resources related to this article, please visit the .**27**Feyereisen, M.; Fitzgerald, G.; Komornicki, A. Use of approximate integrals in ab initio theory. An application in MP2 energy calculations.*Chem. Phys. Lett.*1993,*208*, 359, DOI: 10.1016/0009-2614(93)87156-WGoogle Scholar27https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXkvVSntL8%253D&md5=4d39ebc6228d81fcd03290405a40dbdbUse of approximate integrals in ab initio theory. An application in MP2 energy calculationsFeyereisen, Martin; Fitzgerald, George; Komornicki, AndrewChemical Physics Letters (1993), 208 (5-6), 359-63CODEN: CHPLBC; ISSN:0009-2614.Authors use the resoln. of the identity (RI) as a convenient way to replace the use of four-index two-electron integrals with linear combinations of three-index integrals. The method is broadly applicable to a wide range of problems in quantum chem. Authors demonstrate the effectiveness of RI for the calcn. of MP2 energies. For the water dimer, agreement within 0.1 kcal/mol is obtained with respect to exact MP2 calcns. The RI-MP2 energies require only about 10% of the time required by conventional MP2.**28**Gyevi-Nagy, L.; Kállay, M.; Nagy, P. R. Integral-direct and parallel implementation of the CCSD(T) method: Algorithmic developments and large-scale applications.*J. Chem. Theory Comput.*2020,*16*, 366, DOI: 10.1021/acs.jctc.9b00957Google Scholar28https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BB3MfjvFShsw%253D%253D&md5=62e3430fb75945f63928d737c0cb0d49Integral-Direct and Parallel Implementation of the CCSD(T) Method: Algorithmic Developments and Large-Scale ApplicationsGyevi-Nagy Laszlo; Kallay Mihaly; Nagy Peter RJournal of chemical theory and computation (2020), 16 (1), 366-384 ISSN:.A completely integral-direct, disk I/O, and network traffic economic coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] implementation has been developed relying on the density-fitting approximation. By fully exploiting the permutational symmetry, the presented algorithm is highly operation count and memory-efficient. Our measurements demonstrate excellent strong scaling achieved via hybrid MPI/OpenMP parallelization and a highly competitive, 60-70% utilization of the theoretical peak performance on up to hundreds of cores. The terms whose evaluation time becomes significant only for small- to medium-sized examples have also been extensively optimized. Consequently, high performance is also expected for systems appearing in extensive data sets used, e.g., for density functional or machine learning parametrizations, and in calculations required for certain reduced-cost or local approximations of CCSD(T), such as in our local natural orbital scheme [LNO-CCSD(T)]. The efficiency of this implementation allowed us to perform some of the largest CCSD(T) calculations ever presented for systems of 31-43 atoms and 1037-1569 orbitals using only four to eight many-core CPUs and 1-3 days of wall time. The resulting 13 correlation energies and the 12 corresponding reaction energies and barrier heights are added to our previous benchmark set collecting reference CCSD(T) results of molecules at the applicability limit of current implementations.**29**Almlöf, J. Elimination of energy denominators in Møller-Plesset perturbation theory by a Laplace transform approach.*Chem. Phys. Lett.*1991,*181*, 319, DOI: 10.1016/0009-2614(91)80078-CGoogle Scholar29https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3MXkvF2msL8%253D&md5=7b12bd4b2091509f65775ab277389b0eElimination of energy denominators in Moeller-Plesset perturbation theory by a Laplace transform approachAlmlof, JanChemical Physics Letters (1991), 181 (4), 319-20CODEN: CHPLBC; ISSN:0009-2614.It is shown how the energy denominators encountered in various schemes for electronic structure calcn. can be removed by a Laplace transform technique. The method is applicable to a wide variety of electronic structure calcns.**30**Häser, M.; Almlöf, J. Laplace transform techniques in Møller-Plesset perturbation theory.*J. Chem. Phys.*1992,*96*, 489, DOI: 10.1063/1.462485Google ScholarThere is no corresponding record for this reference.**31**Ayala, P. Y.; Scuseria, G. E. Linear scaling second-order Møller-Plesset theory in the atomic orbital basis for large molecular systems.*J. Chem. Phys.*1999,*110*, 3660, DOI: 10.1063/1.478256Google Scholar31https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXosVWmsQ%253D%253D&md5=1d3a71f19d43f89ec50a315a1d96db18Linear scaling second-order Moeller-Plesset theory in the atomic orbital basis for large molecular systemsAyala, Philippe Y.; Scuseria, Gustavo E.Journal of Chemical Physics (1999), 110 (8), 3660-3671CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We have used Almlof and Haser's Laplace transform idea to eliminate the energy denominator in second-order perturbation theory (MP2) and obtain an energy expression in the AO basis. We show that the asymptotic computational cost of this method scales quadratically with mol. size. We then define AO domains such that selective pairwise interactions can be neglected using well-defined thresholding criteria based on the power law decay properties of the long-range contributions. For large mols., our scheme yields linear scaling computational cost as a function of mol. size. The errors can be controlled in a precise manner and our method reproduces canonical MP2 energies. We present benchmark calcns. of polyglycine chains and water clusters contg. up to 3040 basis functions.**32**Surján, P. R. The MP2 energy as a functional of the Hartree-Fock density matrix.*Chem. Phys. Lett.*2005,*406*, 318– 320, DOI: 10.1016/j.cplett.2005.03.024Google Scholar32https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXjtF2gsbw%253D&md5=375a4f31cb04f375d948e95c7a03cdfbThe MP2 energy as a functional of the Hartree-Fock density matrixSurjan, Peter R.Chemical Physics Letters (2005), 406 (4-6), 318-320CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.)The explicit E[2][P] functional is presented, where E [2] is the second order Moller-Plesset correlation energy and P is the std. Hartree-Fock d. matrix. The ideas leading to this functional are implicit in previous studies, but the significance of its existence has not yet been sufficiently emphasized and its simple explicit form has not been presented. With the proposed functional one may obtain the correlation energy in the absence of MOs, knowing merely the d. matrix. This may further facilitate linear scaling computation of the correlation energy.**33**Kobayashi, M.; Nakai, H. Implementation of Surján’s density matrix formulae for calculating second-order Møller-Plesset energy.*Chem. Phys. Lett.*2006,*420*, 250– 255, DOI: 10.1016/j.cplett.2005.12.088Google Scholar33https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28Xhslyqtrg%253D&md5=89e912a120e1e1d4d979b725e710f2c2Implementation of Surjan's density matrix formulae for calculating second-order Moller-Plesset energyKobayashi, Masato; Nakai, HiromiChemical Physics Letters (2006), 420 (1-3), 250-255CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.)We numerically assess the method for obtaining second-order Moller-Plesset (MP2) energy from the Hartree-Fock d. matrix (DM) recently proposed by Surjan [Surjan, Chem. Phys. Lett. 406 (2005) 318]. It is confirmed that Surjan's method, referred to as DM-Laplace MP2, can obtain MP2 energy accurately by means of appropriate integral quadrature and a matrix exponential evaluation scheme. Numerical tests reveal that the Euler-Maclaurin and the Romberg numerical integration schemes can achieve milli-hartree accuracy with small quadrature points. This Letter also indicates the possibility of the application of DM-Laplace MP2 to linear-scaling SCF techniques, which give approx. DM.**34**Doser, B.; Lambrecht, D. S.; Kussmann, J.; Ochsenfeld, C. Linear-scaling atomic orbital-based second-order Møller-Plesset perturbation theory by rigorous integral screening criteria.*J. Chem. Phys.*2009,*130*, 064107 DOI: 10.1063/1.3072903Google Scholar34https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhvFentbY%253D&md5=5814416cce1e9f2d0e3140008c61358dLinear-scaling atomic orbital-based second-order Moller-Plesset perturbation theory by rigorous integral screening criteriaDoser, Bernd; Lambrecht, Daniel S.; Kussmann, Joerg; Ochsenfeld, ChristianJournal of Chemical Physics (2009), 130 (6), 064107/1-064107/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A Laplace-transformed second-order Moller-Plesset perturbation theory (MP2) method is presented, which allows to achieve linear scaling of the computational effort with mol. size for electronically local structures. Also for systems with a delocalized electronic structure, a cubic or even quadratic scaling behavior is achieved. Numerically significant contributions to the AO (AO)-MP2 energy are preselected using the so-called multipole-based integral ests. (MBIE) introduced earlier by us. Since MBIE provides rigorous upper bounds, numerical accuracy is fully controlled and the exact MP2 result is attained. While the choice of thresholds for a specific accuracy is only weakly dependent upon the mol. system, our AO-MP2 scheme offers the possibility for incremental thresholding: for only little addnl. computational expense, the numerical accuracy can be systematically converged. We illustrate this dependence upon numerical thresholds for the calcn. of intermol. interaction energies for the S22 test set. The efficiency and accuracy of our AO-MP2 method is demonstrated for linear alkanes, stacked DNA base pairs, and carbon nanotubes: e.g., for DNA systems the crossover toward conventional MP2 schemes occurs between one and two base pairs. In this way, it is for the first time possible to compute wave function-based correlation energies for systems contg. more than 1000 atoms with 10 000 basis functions as illustrated for a 16 base pair DNA system on a single-core computer, where no empirical restrictions are introduced and numerical accuracy is fully preserved. (c) 2009 American Institute of Physics.**35**Schäfer, T.; Ramberger, B.; Kresse, G. Quartic scaling MP2 for solids: A highly parallelized algorithm in the plane wave basis.*J. Chem. Phys.*2017,*146*, 104101 DOI: 10.1063/1.4976937Google Scholar35https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC1czms1Knsw%253D%253D&md5=054a349b32abeee80c9412e2aeccf107Quartic scaling MP2 for solids: A highly parallelized algorithm in the plane wave basisSchafer Tobias; Ramberger Benjamin; Kresse GeorgThe Journal of chemical physics (2017), 146 (10), 104101 ISSN:.We present a low-complexity algorithm to calculate the correlation energy of periodic systems in second-order Moller-Plesset (MP2) perturbation theory. In contrast to previous approximation-free MP2 codes, our implementation possesses a quartic scaling, O(N(4)), with respect to the system size N and offers an almost ideal parallelization efficiency. The general issue that the correlation energy converges slowly with the number of basis functions is eased by an internal basis set extrapolation. The key concept to reduce the scaling is to eliminate all summations over virtual orbitals which can be elegantly achieved in the Laplace transformed MP2 formulation using plane wave basis sets and fast Fourier transforms. Analogously, this approach could allow us to calculate second order screened exchange as well as particle-hole ladder diagrams with a similar low complexity. Hence, the presented method can be considered as a step towards systematically improved correlation energies.**36**Pulay, P.; Saebø, S. Orbital-invariant formulation and second-order gradient evaluation in Møller-Plesset perturbation theory.*Theor. Chem. Acc.*1986,*69*, 357, DOI: 10.1007/BF00526697Google Scholar36https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL28XltVakt7g%253D&md5=d16c8d33f0c76e557d97ae63f8cfca22Orbital-invariant formulation and second-order gradient evaluation in Moeller-Plesset perturbation theoryPulay, Peter; Saeboe, SveinTheoretica Chimica Acta (1986), 69 (5-6), 357-68CODEN: TCHAAM; ISSN:0040-5744.Based on the Hylleraas functional form, the second and third orders of the Moeller-Plesset (MP) perturbation theory were reformulated in terms of arbitrary (e.g., localized) internal orbitals, and AOs in the virtual space. The results are strictly equiv. to the canonical formulation if no further approxns. are introduced. The new formalism permits the extension of the local correlation method to MP theory. It also facilitates the treatment of weak pairs at a lower (e.g., second-order) level of theory in CI and coupled-cluster methods. Based on the formalism, an MP2 gradient algorithm is outlined, which does not require the storage of deriv. integrals, integrals with three external MO indexes, and, using the method of N. C. Handy and H. F. Schaefer III (1984), the repeated soln. of the coupled-perturbed SCF equations.**37**Kats, D.; Usvyat, D.; Schütz, M. On the use of the Laplace transform in local correlation methods.*Phys. Chem. Chem. Phys.*2008,*10*, 3430, DOI: 10.1039/b802993hGoogle Scholar37https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXmvVemtLw%253D&md5=5dd80442ae7cda8caa0d66efe739483fOn the use of the Laplace transform in local correlation methodsKats, Danylo; Usvyat, Denis; Schuetz, MartinPhysical Chemistry Chemical Physics (2008), 10 (23), 3430-3439CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)The applicability of the Laplace transform ansatz of Almlof in the context of local correlation methods with a priori restricted sets of wavefunction parameters is explored. A new local MP2 method based on the Laplace transform ansatz is described, its relation to the local MP2 method based on the Pulay ansatz is elucidated, and its accuracy and efficiency are compared to the latter.**38**Nagy, P. R.; Samu, G.; Kállay, M. An integral-direct linear-scaling second-order Møller-Plesset approach.*J. Chem. Theory Comput.*2016,*12*, 4897, DOI: 10.1021/acs.jctc.6b00732Google Scholar38https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhsV2lu7zO&md5=abd0db52ed97c1a4246e976e14b0aca1An Integral-Direct Linear-Scaling Second-Order Moller-Plesset ApproachNagy, Peter R.; Samu, Gyula; Kallay, MihalyJournal of Chemical Theory and Computation (2016), 12 (10), 4897-4914CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)An integral-direct, iteration-free, linear-scaling, local second-order Moller-Plesset (MP2) approach is presented, which is also useful for spin-scaled MP2 calcns. as well as for the efficient evaluation of the perturbative terms of double-hybrid d. functionals. The method is based on a fragmentation approxn.: the correlation contributions of the individual electron pairs are evaluated in domains constructed for the corresponding localized orbitals, and the correlation energies of distant electron pairs are computed with multipole expansions. The required electron repulsion integrals are calcd. directly invoking the d. fitting approxn.; the storage of integrals and intermediates is avoided. The approach also utilizes natural auxiliary functions to reduce the size of the auxiliary basis of the domains and thereby the operation count and memory requirement. Our test calcns. show that the approach recovers 99.9% of the canonical MP2 correlation energy and reproduces reaction energies with an av. (max.) error below 1 kJ/mol (4 kJ/mol). Our benchmark calcns. demonstrate that the new method enables MP2 calcns. for mols. with more than 2300 atoms and 26000 basis functions on a single processor.**39**Saebø, S.*Linear-Scaling Techniques in Computational Chemistry and Physics: Methods and Applications*; Springer: Netherlands, 2011; pp 65– 82.Google ScholarThere is no corresponding record for this reference.**40**Zienau, J.; Clin, L.; Doser, B.; Ochsenfeld, C. Cholesky-decomposed densities in Laplace-based second-order Møller-Plesset perturbation theory.*J. Chem. Phys.*2009,*130*, 204112 DOI: 10.1063/1.3142592Google Scholar40https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXntVSgur0%253D&md5=34625433d59701160948f2dff3305a2aCholesky-decomposed densities in Laplace-based second-order Moller-Plesset perturbation theoryZienau, Jan; Clin, Lucien; Doser, Bernd; Ochsenfeld, ChristianJournal of Chemical Physics (2009), 130 (20), 204112/1-204112/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Based on our linear-scaling AO second-order Moller-Plesset perturbation theory (AO-MP2) method , we explore the use of Cholesky-decompd. pseudodensity (CDD) matrixes within the Laplace formulation. Numerically significant contributions are preselected using our multipole-based integral ests. as upper bounds to two-electron integrals so that the 1/R6 decay behavior of transformed Coulomb-type products is exploited. In addn., we combine our new CDD-MP2 method with the resoln. of the identity (RI) approach. Even though the use of RI results in a method that shows a quadratic scaling behavior in the dominant steps, gains of up to one or two orders of magnitude vs. our original AO-MP2 method are obsd. in particular for larger basis sets. (c) 2009 American Institute of Physics.**41**Maurer, S. A.; Clin, L.; Ochsenfeld, C. Cholesky-decomposed density MP2 with density fitting: Accurate MP2 and double-hybrid DFT energies for large systems.*J. Chem. Phys.*2014,*140*, 224112 DOI: 10.1063/1.4881144Google Scholar41https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXpslCqu7c%253D&md5=9463d1755513305cadf4f01aa16cd2cbCholesky-decomposed density MP2 with density fitting: Accurate MP2 and double-hybrid DFT energies for large systemsMaurer, Simon A.; Clin, Lucien; Ochsenfeld, ChristianJournal of Chemical Physics (2014), 140 (22), 224112/1-224112/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Our recently developed QQR-type integral screening is introduced in our Cholesky-decompd. pseudo-densities Moller-Plesset perturbation theory of second order (CDD-MP2) method. We use the resoln.-of-the-identity (RI) approxn. in combination with efficient integral transformations employing sparse matrix multiplications. The RI-CDD-MP2 method shows an asymptotic cubic scaling behavior with system size and a small prefactor that results in an early crossover to conventional methods for both small and large basis sets. We also explore the use of local fitting approxns. which allow to further reduce the scaling behavior for very large systems. The reliability of our method is demonstrated on test sets for interaction and reaction energies of medium sized systems and on a diverse selection from our own benchmark set for total energies of larger systems. Timings on DNA systems show that fast calcns. for systems with more than 500 atoms are feasible using a single processor core. Parallelization extends the range of accessible system sizes on one computing node with multiple cores to more than 1000 atoms in a double-zeta basis and more than 500 atoms in a triple-zeta basis. (c) 2014 American Institute of Physics.**42**Helmich-Paris, B.; Repisky, M.; Visscher, L. Relativistic Cholesky-decomposed density matrix MP2.*Chem. Phys.*2019,*518*, 38, DOI: 10.1016/j.chemphys.2018.11.009Google Scholar42https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXitlenu7jN&md5=4eeed0769da6e3f6709e0e313aea7514Relativistic Cholesky-decomposed density matrix MP2Helmich-Paris, Benjamin; Repisky, Michal; Visscher, LucasChemical Physics (2019), 518 (), 38-46CODEN: CMPHC2; ISSN:0301-0104. (Elsevier B.V.)We introduce the relativistic Cholesky-decompd. d. (CDD) matrix second-order Moller-Plesset perturbation theory (MP2) energies. The working equations are formulated in terms of the usual intermediates of MP2 when employing the resoln.-of-the-identity approxn. (RI) for two-electron integrals. Those intermediates are obtained by substituting the occupied and virtual quaternion pseudo-d. matrixes of our previously proposed two-component (2C) AO-based MP2 (Helmich-Paris et al., 2016) by the corresponding pivoted quaternion Cholesky factors. While working within the Kramers-restricted formalism, we obtain a formal spin-orbit overhead of 16 and 28 for the Coulomb and exchange contribution to the 2C MP2 correlation energy, resp., compared to a non-relativistic (NR) spin-free CDD-MP2 implementation. This compact quaternion formulation could also be easily explored in any other algorithm to compute the 2C MP2 energy. The quaternion Cholesky factors become sparse for large mols. and, with a block-wise screening, block sparse-matrix multiplication algorithm, we obsd. an effective quadratic scaling of the total wall time for heavy-element contg. linear mols. with increasing system size. The total run time for both NR and 2C calcns. was dominated by the contraction to the exchange energy. We have also investigated a bulky Te-contg. supramol. complex. For such bulky, three-dimensionally extended mols. the present screening scheme has a much larger prefactor and is less effective.**43**Glasbrenner, M.; Graf, D.; Ochsenfeld, C. Efficient Reduced-Scaling Second-Order Møller-Plesset Perturbation Theory with Cholesky-Decomposed Densities and an Attenuated Coulomb Metric.*J. Chem. Theory Comput.*2020,*16*, 6856, DOI: 10.1021/acs.jctc.0c00600Google Scholar43https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXitVykurrM&md5=e1b5deb7744946700ddc855d1c69f960Efficient Reduced-Scaling Second-Order Moller-Plesset Perturbation Theory with Cholesky-Decomposed Densities and an Attenuated Coulomb MetricGlasbrenner, Michael; Graf, Daniel; Ochsenfeld, ChristianJournal of Chemical Theory and Computation (2020), 16 (11), 6856-6868CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a novel, highly efficient method for the computation of second-order Moller-Plesset perturbation theory (MP2) correlation energies, which uses the resoln. of the identity (RI) approxn. and local MOs obtained from a Cholesky decompn. of pseudodensity matrixes (CDD), as in the RI-CDD-MP2 method developed previously in our group [Maurer, S.A. et al., J. Chem. Phys., 2014, 140, 224112]. In addn., we introduce an attenuated Coulomb metric and subsequently redesign the RI-CDD-MP2 method in order to exploit the resulting sparsity in the three-center integrals. Coulomb and exchange energy contributions are computed sep. using specialized algorithms. A simple, yet effective integral screening protocol based on Schwarz ests. is used for the MP2 exchange energy. The Coulomb energy computation and the preceding transformations of the three-center integrals are accelerated using a modified version of the natural blocking approach [Jung, Y., Head-Gordon, M., Phys. Chem. Chem. Phys., 2006, 8, 2831]. Effective subquadratic scaling for a wide range of mol. sizes is demonstrated in test calcns. in conjunction with a low prefactor. The method is shown to enable cost-efficient MP2 calcns. on large mol. systems with several thousand basis functions.**44**Neuhauser, D.; Rabani, E.; Baer, R. Expeditious Stochastic Approach for MP2 Energies in Large Electronic Systems.*J. Chem. Theory Comput.*2013,*9*, 24, DOI: 10.1021/ct300946jGoogle Scholar44https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhvVSqtL7E&md5=00873c7c04cdecfa46df550c78ee1cf7Expeditious Stochastic Approach for MP2 Energies in Large Electronic SystemsNeuhauser, Daniel; Rabani, Eran; Baer, RoiJournal of Chemical Theory and Computation (2013), 9 (1), 24-27CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A fast stochastic method for calcg. the second order Moller-Plesset (MP2) correction to the correlation energy of large systems of electrons is presented. The approach is based on reducing the exact summation over occupied and unoccupied states to a time-dependent trace formula amenable to stochastic sampling. We demonstrate the abilities of the method to treat systems with thousands of electrons using hydrogen passivated silicon spherical nanocrystals represented on a real space grid, much beyond the capabilities of present day MP2 implementations.**45**Willow, S. Y.; Kim, K. S.; Hirata, S. Stochastic evaluation of second-order many-body perturbation energies.*J. Chem. Phys.*2012,*137*, 204122 DOI: 10.1063/1.4768697Google Scholar45https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhslKms77I&md5=2c542905be62e0d3ab11a74fb3e4786aStochastic evaluation of second-order many-body perturbation energiesWillow, Soohaeng Yoo; Kim, Kwang S.; Hirata, SoJournal of Chemical Physics (2012), 137 (20), 204122/1-204122/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)With the aid of the Laplace transform, the canonical expression of the second-order many-body perturbation correction to electronic energy is converted into a sum of two 13-dimensional integrals, the 12-dimensional parts of which are evaluated by Monte Carlo integration. Wt. functions are identified that are anal. normalizable, finite, non-neg. everywhere, and share the same singularities as the integrands. They generate appropriate distributions of four-electron walkers via the Metropolis algorithm, yielding correlation energies of small mols. within a few mEh of the correct values after 108 Monte Carlo steps. This algorithm does away with the integral transformation as the hotspot of the usual algorithms, has a far superior size dependence of cost, does not suffer from the sign problem of some quantum Monte Carlo methods, and potentially easily parallelizable and extensible to other more complex electron-correlation theories. (c) 2012 American Institute of Physics.**46**Barca, G. M. J.; McKenzie, S. C.; Bloomfield, N. J.; Gilbert, A. T. B.; Gill, P. M. W. Q-MP2-OS: Møller-Plesset Correlation Energy by Quadrature.*J. Chem. Theory Comput.*2020,*16*, 1568, DOI: 10.1021/acs.jctc.9b01142Google Scholar46https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhsFKhtrs%253D&md5=5bc60925dc5e5142669a39d0141d02e8Q-MP2-OS: Moller-Plesset Correlation Energy by QuadratureBarca, Giuseppe M. J.; McKenzie, Simon C.; Bloomfield, Nathaniel J.; Gilbert, Andrew T. B.; Gill, Peter M. W.Journal of Chemical Theory and Computation (2020), 16 (3), 1568-1577CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a quadrature-based algorithm for computing the opposite-spin component of the MP2 correlation energy which scales quadratically with basis set size and is well-suited to large-scale parallelization. The key ideas, which are rooted in the earlier work of Hirata and co-workers, are to abandon all two-electron integrals, recast the energy as a seven-dimensional integral, approx. that integral by quadrature, and employ a cutoff strategy to minimize the no. of intermediate quantities. We discuss our implementation in detail and show that it parallelizes almost perfectly on 840 cores for cyclosporine (a mol. with roughly 200 atoms), exhibits scaling for a sequence of polyglycines, and is principally limited by the accuracy of its quadrature.**47**Martínez, T. J.; Carter, E. A. Pseudospectral Møller-Plesset perturbation theory through third order.*J. Chem. Phys.*1994,*100*, 3631, DOI: 10.1063/1.466350Google Scholar47https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXit1Onsrc%253D&md5=54542779da7d2ff27e0d4038bcf4d056Pseudospectral Moeller-Plesset perturbation theory through third orderMartinez, Todd J.; Carter, Emily A.Journal of Chemical Physics (1994), 100 (5), 3631-8CODEN: JCPSA6; ISSN:0021-9606.The authors present a formulation and implementation of Moeller-Plesset perturbation theory in a pseudospectral framework. At the second-order level, the pseudospectral formulation is a formally a factor of N/n faster than conventional approaches, while the third order is formally faster by a factor of n, where N is the no. of AOs and n is the no. of occupied orbitals. The accuracy of the resulting energies is probed for a no. of test cases. Practical timings are presented and show conclusively that the pseudospectral formulation is faster than conventional ones.**48**Kossmann, S.; Neese, F. Efficient Structure Optimization with Second-Order Many-Body Perturbation Theory: The RIJCOSX-MP2 Method.*J. Chem. Theory Comput.*2010,*6*, 2325, DOI: 10.1021/ct100199kGoogle Scholar48https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXot1GisLk%253D&md5=6dd45c3962123232a7e9e5fd36630f8aEfficient Structure Optimization with Second-Order Many-Body Perturbation Theory: The RIJCOSX-MP2 MethodKossmann, Simone; Neese, FrankJournal of Chemical Theory and Computation (2010), 6 (8), 2325-2338CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Efficient energy calcns. and structure optimizations employing second-order Moller-Plesset perturbation theory (MP2) are presented. The application of the RIJCOSX approxn., which involves different approxns. for the formation of the Coulomb- and exchange-type matrixes, to MP2 theory is demonstrated. The RIJCOSX approxn. incorporates the resoln. of the identity' approxn. in terms of a Split-RI-J variant for the evaluation of the Coulomb matrixes and a seminumeric exchange treatment via the chain-of-spheres' algorithm for the formation of the exchange-type matrixes. Beside the derivation of the working equations, the RIJCOSX-MP2 method is benchmarked against the original MP2 and the already highly efficient RI-MP2 method. Energies as well as gradients are computed employing various basis sets and are compared to the conventional MP2 results concerning accuracy and total wall clock times. Speedups of typically a factor of 5-7 in comparison to MP2 can be obsd. for the largest basis set employed in our study. Total energies are reproduced with an av. error of ≤0.8 kcal/mol and min. energy geometries differ by ∼0.1 pm in bond lengths and typically ∼0.2 degrees in bond angles. The RIJCOSX-MP2 gradient parallelizes with a speedup of 8.2 on 10 processors. The algorithms are implemented into the ORCA electronic structure package.**49**Maslen, P. E.; Head-Gordon, M. Non-iterative local second order Møller-Plesset theory.*Chem. Phys. Lett.*1998,*283*, 102, DOI: 10.1016/S0009-2614(97)01333-XGoogle Scholar49https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXhtlCqs7Y%253D&md5=999d3729e8ab38176986d00ffeee0634Non-iterative local second order Moller-Plesset theoryMaslen, P. E.; Head-Gordon, M.Chemical Physics Letters (1998), 283 (1,2), 102-108CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)Second order Moller-Plesset perturbation theory (MP2) is formulated in terms of atom-centered occupied and virtual orbitals. Both the occupied and the virtual orbitals are non-orthogonal. A new parameter-free atoms-in-mols. local approxn. is employed to reduce the cost of the calcn. to cubic scaling, and a quasi-canonical two-particle basis is introduced to enable the soln. of the local MP2 equations via explicit matrix diagonalization rather than iteration.**50**Jung, Y.; Shao, Y.; Head-Gordon, M. Fast evaluation of scaled opposite-spin second-order Møller-Plesset correlation energies using auxiliary basis expansions and exploiting sparsity.*J. Comput. Chem.*2007,*28*, 1953, DOI: 10.1002/jcc.20590Google Scholar50https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXotVynsbk%253D&md5=50644e7112ac163100d2e7daca8e0d99Fast evaluation of scaled opposite spin second-order Moller-Plesset correlation energies using auxiliary basis expansions and exploiting sparsityJung, Yousung; Shao, Yihan; Head-Gordon, MartinJournal of Computational Chemistry (2007), 28 (12), 1953-1964CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)The scaled opposite spin Moller-Plesset method (SOS-MP2) is an economical way of obtaining correlation energies that are computationally cheaper, and yet, in a statistical sense, of higher quality than std. MP2 theory, by introducing one empirical parameter. But SOS-MP2 still has a fourth-order scaling step that makes the method inapplicable to very large mol. systems. We reduce the scaling of SOS-MP2 by exploiting the sparsity of expansion coeffs. and local integral matrixes, by performing local auxiliary basis expansions for the occupied-virtual product distributions. To exploit sparsity of 3-index local quantities, we use a blocking scheme in which entire zero-rows and columns, for a given third global index, are deleted by comparison against a numerical threshold. This approach minimizes sparse matrix book-keeping overhead, and also provides sufficiently large submatrixes after blocking, to allow efficient matrix-matrix multiplies. The resulting algorithm is formally cubic scaling, and requires only moderate computational resources (quadratic memory and disk space) and, in favorable cases, is shown to yield effective quadratic scaling behavior in the size regime we can apply it to. Errors assocd. with local fitting using the attenuated Coulomb metric and numerical thresholds in the blocking procedure are found to be insignificant in terms of the predicted relative energies. A diverse set of test calcns. shows that the size of system where significant computational savings can be achieved depends strongly on the dimensionality of the system, and the extent of localizability of the MOs.**51**Förster, A.; Franchini, M.; van Lenthe, E.; Visscher, L. A Quadratic Pair Atomic Resolution of the Identity Based SOS-AO-MP2 Algorithm Using Slater Type Orbitals.*J. Chem. Theory Comput.*2020,*16*, 875, DOI: 10.1021/acs.jctc.9b00854Google Scholar51https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BB3MbpsVCgsA%253D%253D&md5=f19110433d706d7ce8a95981b55a3573A Quadratic Pair Atomic Resolution of the Identity Based SOS-AO-MP2 Algorithm Using Slater Type OrbitalsForster Arno; Franchini Mirko; Visscher Lucas; Franchini Mirko; van Lenthe ErikJournal of chemical theory and computation (2020), 16 (2), 875-891 ISSN:.We report a production level implementation of pair atomic resolution of the identity (PARI) based second-order Moller-Plesset perturbation theory (MP2) in the Slater type orbital (STO) based Amsterdam Density Functional (ADF) code. As demonstrated by systematic benchmarks, dimerization and isomerization energies obtained with our code using STO basis sets of triple-ζ-quality show mean absolute deviations from Gaussian type orbital, canonical, basis set limit extrapolated, global density fitting (DF)-MP2 results of less than 1 kcal/mol. Furthermore, we introduce a quadratic scaling atomic orbital based spin-opposite-scaled (SOS)-MP2 approach with a very small prefactor. Due to a worst-case scaling of [Formula: see text], our implementation is very fast already for small systems and shows an exceptionally early crossover to canonical SOS-PARI-MP2. We report computational wall time results for linear as well as for realistic three-dimensional molecules and show that triple-ζ quality calculations on molecules of several hundreds of atoms are only a matter of a few hours on a single compute node, the bottleneck of the computations being the SCF rather than the post-SCF energy correction.**52**Förster, A.; Visscher, L. Double hybrid DFT calculations with Slater type orbitals.*J. Comput. Chem.*2020,*41*, 1660, DOI: 10.1002/jcc.26209Google Scholar52https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BB38zlvFGrtw%253D%253D&md5=df9dd9cc70a8109012fa4f17204467f6Double hybrid DFT calculations with Slater type orbitalsForster Arno; Visscher LucasJournal of computational chemistry (2020), 41 (18), 1660-1684 ISSN:.On a comprehensive database with 1,644 datapoints, covering several aspects of main-group as well as of transition metal chemistry, we assess the performance of 60 density functional approximations (DFA), among them 36 double hybrids (DH). All calculations are performed using a Slater type orbital (STO) basis set of triple-ζ (TZ) quality and the highly efficient pair atomic resolution of the identity approach for the exchange- and Coulomb-term of the KS matrix (PARI-K and PARI-J, respectively) and for the evaluation of the MP2 energy correction (PARI-MP2). Employing the quadratic scaling SOS-AO-PARI-MP2 algorithm, DHs based on the spin-opposite-scaled (SOS) MP2 approximation are benchmarked against a database of large molecules. We evaluate the accuracy of STO/PARI calculations for B3LYP as well as for the DH B2GP-PLYP and show that the combined basis set and PARI-error is comparable to the one obtained using the well-known def2-TZVPP Gaussian-type basis set in conjunction with global density fitting. While quadruple-ζ (QZ) calculations are currently not feasible for PARI-MP2 due to numerical issues, we show that, on the TZ level, Jacob's ladder for classifying DFAs is reproduced. However, while the best DHs are more accurate than the best hybrids, the improvements are less pronounced than the ones commonly found on the QZ level. For conformers of organic molecules and noncovalent interactions where very high accuracy is required for qualitatively correct results, DHs provide only small improvements over hybrids, while they still excel in thermochemistry, kinetics, transition metal chemistry and the description of strained organic systems.**53**Hohenstein, E. G.; Parrish, R. M.; Martínez, T. J. Tensor hypercontraction density fitting. I. Quartic scaling second- and third-order Møller-Plesset perturbation theory.*J. Chem. Phys.*2012,*137*, 044103 DOI: 10.1063/1.4732310Google Scholar53https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhtVOgtb7M&md5=5ce6ce5cd9f7915a7d02ed5bc5fed4f2Tensor hypercontraction density fitting. I. Quartic scaling second- and third-order Moller-Plesset perturbation theoryHohenstein, Edward G.; Parrish, Robert M.; Martinez, Todd J.Journal of Chemical Physics (2012), 137 (4), 044103/1-044103/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Many approxns. have been developed to help deal with the O(N4) growth of the electron repulsion integral (ERI) tensor, where N is the no. of one-electron basis functions used to represent the electronic wavefunction. Of these, the d. fitting (DF) approxn. is currently the most widely used despite the fact that it is often incapable of altering the underlying scaling of computational effort with respect to mol. size. We present a method for exploiting sparsity in three-center overlap integrals through tensor decompn. to obtain a low-rank approxn. to d. fitting (tensor hypercontraction d. fitting or THC-DF). This new approxn. reduces the 4th-order ERI tensor to a product of five matrixes, simultaneously reducing the storage requirement as well as increasing the flexibility to regroup terms and reduce scaling behavior. As an example, we demonstrate such a scaling redn. for second- and third-order perturbation theory (MP2 and MP3), showing that both can be carried out in O(N4) operations. This should be compared to the usual scaling behavior of O(N5) and O(N6) for MP2 and MP3, resp. The THC-DF technique can also be applied to other methods in electronic structure theory, such as coupled-cluster and CI, promising significant gains in computational efficiency and storage redn. (c) 2012 American Institute of Physics.**54**Bangerter, F. H.; Glasbrenner, M.; Ochsenfeld, C. Low-Scaling Tensor Hypercontraction in the Cholesky Molecular Orbital Basis Applied to Second-Order Møller-Plesset Perturbation Theory.*J. Chem. Theory Comput.*2021,*17*, 211, DOI: 10.1021/acs.jctc.0c00934Google Scholar54https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXis12rtLrN&md5=2a77224fa989a3f704c8aca79c12d584Low-Scaling Tensor Hypercontraction in the Cholesky Molecular Orbital Basis Applied to Second-Order Moller-Plesset Perturbation TheoryBangerter, Felix H.; Glasbrenner, Michael; Ochsenfeld, ChristianJournal of Chemical Theory and Computation (2021), 17 (1), 211-221CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We employ various reduced scaling techniques to accelerate the recently developed least-squares tensor hypercontraction (LS-THC) approxn. [Parrish, R.M. et al., J. Chem. Phys. 2012, 137, 224106] for electron repulsion integrals (ERIs) and apply it to second-order Moller-Plesset perturbation theory (MP2). The grid-projected ERI tensors are efficiently constructed using a localized Cholesky MO basis from d.-fitted integrals with an attenuated Coulomb metric. Addnl., rigorous integral screening and the natural blocking matrix format are applied to reduce the complexity of this step. By recasting the equations to form the quantized representation of the 1/r operator Z into the form of a system of linear equations, the bottleneck of inverting the grid metric via pseudoinversion is removed. This leads to a reduced scaling THC algorithm and application to MP2 yields the (sub-)quadratically scaling THC-ω-RI-CDD-SOS-MP2 method. The efficiency of this method is assessed for various systems including DNA fragments with over 8000 basis functions and the subquadratic scaling is illustrated.**55**Del Ben, M.; Hutter, J.; VandeVondele, J. Second-Order Møller-Plesset Perturbation Theory in the Condensed Phase: An Efficient and Massively Parallel Gaussian and Plane Waves Approach.*J. Chem. Theory Comput.*2012,*8*, 4177, DOI: 10.1021/ct300531wGoogle Scholar55https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38Xht12isL3F&md5=90469d4482aae04ee25d1fe5de6e8621Second-Order Moller-Plesset Perturbation Theory in the Condensed Phase: An Efficient and Massively Parallel Gaussian and Plane Waves ApproachDel Ben, Mauro; Hutter, Jurg; VandeVondele, JoostJournal of Chemical Theory and Computation (2012), 8 (11), 4177-4188CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A novel algorithm, based on a hybrid Gaussian and plane waves (GPW) approach, is developed for the canonical second-order Moller-Plesset perturbation energy (MP2) of finite and extended systems. The key aspect of the method is that the electron repulsion integrals (ia|λσ) are computed by direct integration between the products of Gaussian basis functions λσ and the electrostatic potential arising from a given occupied-virtual pair d. ia. The electrostatic potential is obtained in a plane waves basis set after solving the Poisson equation in Fourier space. In particular, for condensed phase systems, this scheme is highly efficient. Furthermore, our implementation has low memory requirements and displays excellent parallel scalability up to 100 000 processes. In this way, canonical MP2 calcns. for condensed phase systems contg. hundreds of atoms or more than 5000 basis functions can be performed within minutes, while systems up to 1000 atoms and 10 000 basis functions remain feasible. Solid LiH has been employed as a benchmark to study basis set and system size convergence. Lattice consts. and cohesive energies of various mol. crystals have been studied with MP2 and double-hybrid functionals.**56**Katouda, M.; Naruse, A.; Hirano, Y.; Nakajima, T. Massively parallel algorithm and implementation of RI-MP2 energy calculation for peta-scale many-core supercomputers.*J. Comput. Chem.*2016,*37*, 2623, DOI: 10.1002/jcc.24491Google Scholar56https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhsFamsbfN&md5=0f27c2f6caea947b50e1a4323da02dddMassively parallel algorithm and implementation of RI-MP2 energy calculation for peta-scale many-core supercomputersKatouda, Michio; Naruse, Akira; Hirano, Yukihiko; Nakajima, TakahitoJournal of Computational Chemistry (2016), 37 (30), 2623-2633CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)A new parallel algorithm and its implementation for the RI-MP2 energy calcn. utilizing peta-flop-class many-core supercomputers are presented. Some improvements from the previous algorithm have been performed: (1) a dual-level hierarchical parallelization scheme that enables the use of more than 10,000 Message Passing Interface (MPI) processes and (2) a new data communication scheme that reduces network communication overhead. A multi-node and multi-GPU implementation of the present algorithm is presented for calcns. on a central processing unit (CPU)/graphics processing unit (GPU) hybrid supercomputer. Benchmark results of the new algorithm and its implementation using the K computer (CPU clustering system) and TSUBAME 2.5 (CPU/GPU hybrid system) demonstrate high efficiency. The peak performance of 3.1 PFLOPS is attained using 80,199 nodes of the K computer. The peak performance of the multi-node and multi-GPU implementation is 514 TFLOPS using 1349 nodes and 4047 GPUs of TSUBAME 2.5.**57**Zaleśny, R.; Papadopoulos, M.; Mezey, P.; Leszczynski, J.*Linear-Scaling Techniques in Computational Chemistry and Physics: Methods and Applications*; Challenges and Advances in Computational Chemistry and Physics; Springer: Netherlands, 2011.Google ScholarThere is no corresponding record for this reference.**58**Herbert, J. M. Fantasy versus reality in fragment-based quantum chemistry.*J. Chem. Phys.*2019,*151*, 170901 DOI: 10.1063/1.5126216Google Scholar58https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXitFWgtbjP&md5=2a51bbcdde3d5703657e5c22f2f13659Fantasy versus reality in fragment-based quantum chemistryHerbert, John M.Journal of Chemical Physics (2019), 151 (17), 170901/1-170901/38CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A review. Since the introduction of the fragment MO method 20 years ago, fragment-based approaches have occupied a small but growing niche in quantum chem. These methods decomp. a large mol. system into subsystems small enough to be amenable to electronic structure calcns., following which the subsystem information is reassembled in order to approx. an otherwise intractable supersystem calcn. Fragmentation sidesteps the steep rise (with respect to system size) in the cost of ab initio calcns., replacing it with a distributed cost across numerous computer processors. Such methods are attractive, in part, because they are easily parallelizable and therefore readily amenable to exascale computing. As such, there has been hope that distributed computing might offer the proverbial "free lunch" in quantum chem., with the entre´e being high-level calcns. on very large systems. While fragment-based quantum chem. can count many success stories, there also exists a seedy underbelly of rarely acknowledged problems. As these methods begin to mature, it is time to have a serious conversation about what they can and cannot be expected to accomplish in the near future. Both successes and challenges are highlighted in this Perspective. (c) 2019 American Institute of Physics.**59***Fragmentation: Toward Accurate Calculations on Complex Molecular Systems*; Gordon, M., Ed.; Wiley: New York, 2017.Google ScholarThere is no corresponding record for this reference.**60**Pulay, P. Localizability of dynamic electron correlation.*Chem. Phys. Lett.*1983,*100*, 151, DOI: 10.1016/0009-2614(83)80703-9Google Scholar60https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL3sXls1Chsro%253D&md5=99921f8590340e93532de025c24af827Localizability of dynamic electron correlationPulay, PeterChemical Physics Letters (1983), 100 (2), 151-4CODEN: CHPLBC; ISSN:0009-2614.The convergence of the intrapair correlation energy for a localized internal orbital was studied as the virtual subspace was enlarged. At variance with previous investigation of this kind, the virtual subspace was represented in AO's, which allowed definition of the spatial relations between the orbitals involved. Typically, over 98% of the pair correlation energy was recovered with a small local basis set consisting of the valence orbitals of the atoms with which the electron pair was assocd. This opens the possibility of an efficient CI procedure based on localized pairs. The mols. considered are H2O2, butadiene, and propane.**61**Kurashige, Y.; Yang, J.; Chan, G. K.-L.; Manby, F. R. Optimization of orbital-specific virtuals in local Møller-Plesset perturbation theory.*J. Chem. Phys.*2012,*136*, 124106 DOI: 10.1063/1.3696962Google Scholar61https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XksFentL0%253D&md5=ce9b935c12aad93c7ace5410561688c9Optimization of orbital-specific virtuals in local Moller-Plesset perturbation theoryKurashige, Yuki; Yang, Jun; Chan, Garnet K.-L.; Manby, Frederick R.Journal of Chemical Physics (2012), 136 (12), 124106/1-124106/7CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present an orbital-optimized version of our orbital-specific-virtuals second-order Moller-Plesset perturbation theory (OSV-MP2). The OSV model is a local correlation ansatz with a small basis of virtual functions for each occupied orbital. It is related to the Pulay-Saebo approach, in which domains of virtual orbitals are drawn from a single set of projected AOs; but here the virtual functions assocd. with a particular occupied orbital are specifically tailored to the correlation effects in which that orbital participates. In this study, the shapes of the OSVs are optimized simultaneously with the OSV-MP2 amplitudes by minimizing the Hylleraas functional or approxns. to it. It is found that optimized OSVs are considerably more accurate than the OSVs obtained through singular value decompn. of diagonal blocks of MP2 amplitudes, as used in our earlier work. Orbital-optimized OSV-MP2 recovers smooth potential energy surfaces regardless of the no. of virtuals. Full optimization is still computationally demanding, but orbital optimization in a diagonal or Kapuy-type MP2 approxn. provides an attractive scheme for detg. accurate OSVs. (c) 2012 American Institute of Physics.**62**Riplinger, C.; Neese, F. An efficient and near linear scaling pair natural orbital based local coupled cluster method.*J. Chem. Phys.*2013,*138*, 034106 DOI: 10.1063/1.4773581Google Scholar62https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXpslOqtw%253D%253D&md5=4327115b95524107245acb44ff4aaa7bAn efficient and near linear scaling pair natural orbital based local coupled cluster methodRiplinger, Christoph; Neese, FrankJournal of Chemical Physics (2013), 138 (3), 034106/1-034106/18CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)In previous publications, it was shown that an efficient local coupled cluster method with single- and double excitations can be based on the concept of pair natural orbitals (PNOs) . The resulting local pair natural orbital-coupled-cluster single double (LPNO-CCSD) method has since been proven to be highly reliable and efficient. For large mols., the no. of amplitudes to be detd. is reduced by a factor of 105-106 relative to a canonical CCSD calcn. on the same system with the same basis set. In the original method, the PNOs were expanded in the set of canonical virtual orbitals and single excitations were not truncated. This led to a no. of fifth order scaling steps that eventually rendered the method computationally expensive for large mols. (e.g., >100 atoms). In the present work, these limitations are overcome by a complete redesign of the LPNO-CCSD method. The new method is based on the combination of the concepts of PNOs and projected AOs (PAOs). Thus, each PNO is expanded in a set of PAOs that in turn belong to a given electron pair specific domain. In this way, it is possible to fully exploit locality while maintaining the extremely high compactness of the original LPNO-CCSD wavefunction. No terms are dropped from the CCSD equations and domains are chosen conservatively. The correlation energy loss due to the domains remains below <0.05%, which implies typically 15-20 but occasionally up to 30 atoms per domain on av. The new method has been given the acronym DLPNO-CCSD ("domain based LPNO-CCSD"). The method is nearly linear scaling with respect to system size. The original LPNO-CCSD method had three adjustable truncation thresholds that were chosen conservatively and do not need to be changed for actual applications. In the present treatment, no addnl. truncation parameters have been introduced. Any addnl. truncation is performed on the basis of the three original thresholds. There are no real-space cutoffs. Single excitations are truncated using singles-specific natural orbitals. Pairs are prescreened according to a multipole expansion of a pair correlation energy est. based on local orbital specific virtual orbitals (LOSVs). Like its LPNO-CCSD predecessor, the method is completely of black box character and does not require any user adjustments. It is shown here that DLPNO-CCSD is as accurate as LPNO-CCSD while leading to computational savings exceeding one order of magnitude for larger systems. The largest calcns. reported here featured >8800 basis functions and >450 atoms. In all larger test calcns. done so far, the LPNO-CCSD step took less time than the preceding Hartree-Fock calcn., provided no approxns. have been introduced in the latter. Thus, based on the present development reliable CCSD calcns. on large mols. with unprecedented efficiency and accuracy are realized. (c) 2013 American Institute of Physics.**63**Saitow, M.; Becker, U.; Riplinger, C.; Valeev, E. F.; Neese, F. A new near-linear scaling, efficient and accurate, open-shell domain-based local pair natural orbital coupled cluster singles and doubles theory.*J. Chem. Phys.*2017,*146*, 164105 DOI: 10.1063/1.4981521Google Scholar63https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXmvVeqsL8%253D&md5=898703521d990dfd299c935e34adbfa6A new near-linear scaling, efficient and accurate, open-shell domain-based local pair natural orbital coupled cluster singles and doubles theorySaitow, Masaaki; Becker, Ute; Riplinger, Christoph; Valeev, Edward F.; Neese, FrankJournal of Chemical Physics (2017), 146 (16), 164105/1-164105/31CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The Coupled-Cluster expansion, truncated after single and double excitations (CCSD), provides accurate and reliable mol. electronic wave functions and energies for many mol. systems around their equil. geometries. However, the high computational cost, which is well-known to scale as O(N6) with system size N, has limited its practical application to small systems consisting of not more than approx. 20-30 atoms. To overcome these limitations, low-order scaling approxns. to CCSD have been intensively investigated over the past few years. In our previous work, we have shown that by combining the pair natural orbital (PNO) approach and the concept of orbital domains it is possible to achieve fully linear scaling CC implementations (DLPNO-CCSD and DLPNO-CCSD(T)) that recover around 99.9% of the total correlation energy [C. Riplinger et al., J. Chem. Phys. 144, 024109 (2016)]. The prodn. level implementations of the DLPNO-CCSD and DLPNO-CCSD(T) methods were shown to be applicable to realistic systems composed of a few hundred atoms in a routine, black-box fashion on relatively modest hardware. In 2011, a reduced-scaling CCSD approach for high-spin open-shell UHF ref. wave functions was proposed (UHF-LPNO-CCSD) [A. Hansen et al., J. Chem. Phys. 135, 214102 (2011)]. After a few years of experience with this method, a few shortcomings of UHF-LPNO-CCSD were noticed that required a redesign of the method, which is the subject of this paper. To this end, we employ the high-spin open-shell variant of the N-electron valence perturbation theory formalism to define the initial guess wave function, and consequently also the open-shell PNOs. The new PNO ansatz properly converges to the closed-shell limit since all truncations and approxns. have been made in strict analogy to the closed-shell case. Furthermore, given the fact that the formalism uses a single set of orbitals, only a single PNO integral transformation is necessary, which offers large computational savings. We show that, with the default PNO truncation parameters, approx. 99.9% of the total CCSD correlation energy is recovered for open-shell species, which is comparable to the performance of the method for closed-shells. UHF-DLPNO-CCSD shows a linear scaling behavior for closed-shell systems, while linear to quadratic scaling is obtained for open-shell systems. The largest systems we have considered contain more than 500 atoms and feature more than 10 000 basis functions with a triple-ζ quality basis set. (c) 2017 American Institute of Physics.**64**Ma, Q.; Werner, H.-J. Explicitly correlated local coupled-cluster methods using pair natural orbitals.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2018,*8*, e1371, DOI: 10.1002/wcms.1371Google ScholarThere is no corresponding record for this reference.**65**Krause, C.; Werner, H.-J. Scalable Electron Correlation Methods. 6. Local Spin-Restricted Open-Shell Second-Order Møller-Plesset Perturbation Theory Using Pair Natural Orbitals: PNO-RMP2.*J. Chem. Theory Comput.*2019,*15*, 987, DOI: 10.1021/acs.jctc.8b01012Google ScholarThere is no corresponding record for this reference.https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=&md5=9874b665cc7a056b8e2f928dd3112440**66**Hättig, C.; Tew, D. P.; Helmich, B. Local explicitly correlated second- and third-order Møller-Plesset perturbation theory with pair natural orbitals.*J. Chem. Phys.*2012,*136*, 204105 DOI: 10.1063/1.4719981Google Scholar66https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC38fjsFWjtA%253D%253D&md5=6f9ac0ad9fc79787cfc422c94df07c55Local explicitly correlated second- and third-order Moller-Plesset perturbation theory with pair natural orbitalsHattig Christof; Tew David P; Helmich BenjaminThe Journal of chemical physics (2012), 136 (20), 204105 ISSN:.We present an algorithm for computing explicitly correlated second- and third-order Moller-Plesset energies near the basis set limit for large molecules with a cost that scales formally as N(4) with system size N. This is achieved through a hybrid approach where locality is exploited first through orbital specific virtuals (OSVs) and subsequently through pair natural orbitals (PNOs) and integrals are approximated using density fitting. Our method combines the low orbital transformation costs of the OSVs with the compactness of the PNO representation of the doubles amplitude vector. The N(4) scaling does not rely upon the a priori definition of domains, enforced truncation of pair lists, or even screening and the energies converge smoothly to the canonical values with decreasing occupation number thresholds, used in the selection of the PNO basis. For MP2.5 intermolecular interaction energies, we find that 99% of benchmark basis set limit correlation energy contributions are recovered using an aug-cc-pVTZ basis and that on average only 50 PNOs are required to correlate the significant orbital pairs.**67**Schmitz, G.; Hättig, C. Perturbative triples correction for local pair natural orbital based explicitly correlated CCSD(F12*) using Laplace transformation techniques.*J. Chem. Phys.*2016,*145*, 234107 DOI: 10.1063/1.4972001Google Scholar67https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XitFamur%252FM&md5=40354d9b70054019928647f19f455b8fPerturbative triples correction for local pair natural orbital based explicitly correlated CCSD(F12*) using Laplace transformation techniquesSchmitz, Gunnar; Haettig, ChristofJournal of Chemical Physics (2016), 145 (23), 234107/1-234107/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present an implementation of pair natural orbital coupled cluster singles and doubles with perturbative triples, PNO-CCSD(T), which avoids the quasi-canonical triples approxn. (T0) where couplings due to off-diagonal Fock matrix elements are neglected. A numerical Laplace transformation of the canonical expression for the perturbative (T) triples correction is used to avoid an I/O and storage bottleneck for the triples amplitudes. Results for a test set of reaction energies show that only very few Laplace grid points are needed to obtain converged energy differences and that PNO-CCSD(T) is a more robust approxn. than PNO-CCSD(T0) with a reduced mean abs. deviation from canonical CCSD(T) results. We combine the PNO-based (T) triples correction with the explicitly correlated PNO-CCSD(F12*) method and investigate the use of specialized F12-PNOs in the conventional triples correction. We find that no significant addnl. errors are introduced and that PNO-CCSD(F12*)(T) can be applied in a black box manner. (c) 2016 American Institute of Physics.**68**Collins, M. A.; Bettens, R. P. A. Energy-Based Molecular Fragmentation Methods.*Chem. Rev.*2015,*115*, 5607, DOI: 10.1021/cr500455bGoogle Scholar68https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXmtVajsbY%253D&md5=a83ac50604af530b6d89c94f1a6b6df6Energy-Based Molecular Fragmentation MethodsCollins, Michael A.; Bettens, Ryan P. A.Chemical Reviews (Washington, DC, United States) (2015), 115 (12), 5607-5642CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)A review including the following topics: methods and principles, applications and examples, and speculations and future developments etc.**69**Raghavachari, K.; Saha, A. Accurate Composite and Fragment-Based Quantum Chemical Models for Large Molecules.*Chem. Rev.*2015,*115*, 5643– 5677, DOI: 10.1021/cr500606eGoogle Scholar69https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXmtVajs7o%253D&md5=b04535888d42fe460c4748e86d8a6363Accurate Composite and Fragment-Based Quantum Chemical Models for Large MoleculesRaghavachari, Krishnan; Saha, ArjunChemical Reviews (Washington, DC, United States) (2015), 115 (12), 5643-5677CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)A review. A range of composite methods have been reviewed with the ultimate goal of obtaining accurate energies on large mols. Direct calcns. are possible on small mols. using extrapolated coupled cluster approaches to obtain results within chem. accuracy. Medium-sized mols. can be treated with composite models such as Gn. Error-cancellation strategies are discussed for larger mols. using a hierarchy of chem. based ideas. Finally, a variety of fragment-based methods are discussed as important tools to remove the steep computational bottleneck for large mols. A generalized view of classification of all the major fragment-based methods has also been provided. The potential of fragment-based methods in understanding complex chem. and phys. phenomena in large mols., like DNA, proteins, clusters, and crystals, can be definitely seen. With the combination of new methods, algorithms, and rapid developments in high performance computing, fragment-based quantum chem. clearly will have high impact in the next decade.**70**Friedrich, J.; Dolg, M. Fully Automated Incremental Evaluation of MP2 and CCSD(T) Energies: Application to Water Clusters.*J. Chem. Theory Comput.*2009,*5*, 287, DOI: 10.1021/ct800355eGoogle Scholar70https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXlvFygsg%253D%253D&md5=b899f8700bd63772edc29fd38ff91363Fully Automated Incremental Evaluation of MP2 and CCSD(T) Energies: Application to Water ClustersFriedrich, Joachim; Dolg, MichaelJournal of Chemical Theory and Computation (2009), 5 (2), 287-294CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A fully automated implementation of the incremental scheme for CCSD energies has been extended to treat MP2 and CCSD(T) energies. It is shown in applications on water clusters that the error of the incremental expansion for the total energy is below 1 kcal/mol already at second or third order. It is demonstrated that the approach saves CPU time, RAM, and disk space. Finally it is shown that the calcns. can be run in parallel on up to 50 CPUs, without significant loss of computer time.**71**Fiedler, B.; Schmitz, G.; Hättig, C.; Friedrich, J. Combining Accuracy and Efficiency: An Incremental Focal-Point Method Based on Pair Natural Orbitals.*J. Chem. Theory Comput.*2017,*13*, 6023, DOI: 10.1021/acs.jctc.7b00654Google Scholar71https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhs1yhsLvE&md5=bd8c436e8cbf5d39b1a0c3a4470689dcCombining Accuracy and Efficiency: An Incremental Focal-Point Method Based on Pair Natural OrbitalsFiedler, Benjamin; Schmitz, Gunnar; Haettig, Christof; Friedrich, JoachimJournal of Chemical Theory and Computation (2017), 13 (12), 6023-6042CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)In this work we present a new PNO-based incremental scheme to calc. CCSD(T) and CCSD(T0) reaction, interaction and binding energies. We perform an extensive anal., which shows small incremental errors similar to previous non-PNO calcns. Furthermore, slight PNO errors are obtained by using TPNO = TTNO with appropriate values of 10-7 to 10-8 for reactions and 10-8 for interaction or binding energies. The combination with the efficient MP2 focal-point approach yields chem. accuracy relative to the complete basis-set (CBS) limit. In this method small basis sets (cc-pVDZ, def2-TZVP) for the CCSD(T) part are sufficient in case of reactions or interactions, while some larger ones (e.g. (aug)-cc-pVTZ) are necessary for mol. clusters. For these larger basis sets we show the very high efficiency of our scheme. We obtain not only tremendous decreases of the wall times (i.e. factors > 102) due to the parallelization of the increment calcns. as well as of the total times due to the application of PNOs (i.e. compared to the normal incremental scheme) but also smaller total times with respect to the std. PNO method. That way, our new method features a perfect applicability by combining an excellent accuracy with a very high efficiency as well as the accessibility to larger systems due to the sepn. of the full computation into several small increments.**72**Kobayashi, M.; Nakai, H. Divide-and-conquer-based linear-scaling approach for traditional and renormalized coupled cluster methods with single, double, and noniterative triple excitations.*J. Chem. Phys.*2009,*131*, 114108 DOI: 10.1063/1.3211119Google Scholar72https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXht1alsrjE&md5=ba9885e67d12231f3c5f6fa1373397e9Divide-and-conquer-based linear-scaling approach for traditional and renormalized coupled cluster methods with single, double, and noniterative triple excitationsKobayashi, Masato; Nakai, HiromiJournal of Chemical Physics (2009), 131 (11), 114108/1-114108/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We have reported the divide-and-conquer (DC)-based linear-scaling correlation treatment of coupled-cluster method with single and double excitations (CCSD). In the DC-CCSD method, the CCSD equations derived from subsystem orbitals are solved for each subsystem in order to obtain the total correlation energy by summing up subsystem contributions using energy d. anal. In this study, we extend the DC-CCSD method for treating noniterative perturbative triple excitations using CCSD T1 and T2 amplitudes, namely, CCSD(T). In the DC-CCSD(T) method, the so-called (T) corrections are also computed for each subsystem. Numerical assessments indicate that DC-CCSD(T) reproduces the CCSD(T) results with high accuracy and significantly less computational cost. We further extend the DC-based correlation method to renormalized CCSD(T) for avoiding the divergence that occurs in multireference problems such as bond dissocn. (c) 2009 American Institute of Physics.**73**Nakano, M.; Yoshikawa, T.; Hirata, S.; Seino, J.; Nakai, H. Computerized implementation of higher-order electron-correlation methods and their linear-scaling divide-and-conquer extensions.*J. Comput. Chem.*2017,*38*, 2520– 2527, DOI: 10.1002/jcc.24912Google Scholar73https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhtlSls7jF&md5=583a90f11f8e55f8d04305189c1b57f1Computerized implementation of higher-order electron-correlation methods and their linear-scaling divide-and-conquer extensionsNakano, Masahiko; Yoshikawa, Takeshi; Hirata, So; Seino, Junji; Nakai, HiromiJournal of Computational Chemistry (2017), 38 (29), 2520-2527CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)We have implemented a linear-scaling divide-and-conquer (DC)-based higher-order coupled-cluster (CC) and Moller-Plesset perturbation theories (MPPT) as well as their combinations automatically by means of the tensor contraction engine, which is a computerized symbolic algebra system. The DC-based energy expressions of the std. CC and MPPT methods and the CC methods augmented with a perturbation correction were proposed for up to high excitation orders [e.g., CCSDTQ, MP4, and CCSD(2)TQ]. The numerical assessment for hydrogen halide chains, polyene chains, and first coordination sphere (C1) model of photoactive yellow protein has revealed that the DC-based correlation methods provide reliable correlation energies with significantly less computational cost than that of the conventional implementations. © 2017 Wiley Periodicals, Inc.**74**Mochizuki, Y.; Yamashita, K.; Nakano, T.; Okiyama, Y.; Fukuzawa, K.; Taguchi, N.; Tanaka, S. Higher-order correlated calculations based on fragment molecular orbital scheme.*Theor. Chem. Acc.*2011,*130*, 515– 530, DOI: 10.1007/s00214-011-1036-3Google Scholar74https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhsV2mu7%252FJ&md5=ffe0598673ad6ec190cf3e560c32dde0Higher-order correlated calculations based on fragment molecular orbital schemeMochizuki, Yuji; Yamashita, Katsumi; Nakano, Tatsuya; Okiyama, Yoshio; Fukuzawa, Kaori; Taguchi, Naoki; Tanaka, ShigenoriTheoretical Chemistry Accounts (2011), 130 (2-3), 515-530CODEN: TCACFW; ISSN:1432-2234. (Springer)We have developed a new module for higher-order correlated methods up to coupled-cluster singles and doubles with perturbative triples (CCSD(T)). The matrix-matrix operations through the routine were pursued for a no. of contractions. This code was then incorporated into the program for the fragment MO (FMO) calcns. Intra-fragment processings were parallelized with OpenMP in a node-wise fashion, whereas the message passing interface (MPI) was used for the fragment-wise parallelization over nodes. Our new implementation made the FMO-based higher-order calcns. applicable to realistic proteins. We have performed several benchmark tests on the Earth Simulator (ES2), a massively parallel computer. For example, the FMO-CCSD(T)/6-31G job for the HIV-1 protease (198 amino acid residues)-lopinavir complex was completed in 9.8 h with 512 processors (or 64 nodes). Another example was the influenza neuraminidase (386 residues) with oseltamivir calcd. at the full fourth-order Moller-Plesset perturbation level (MP4), of which job timing was 10.3 h with 1024 processors. The applicability of the methods to commodity cluster computers was tested as well.**75**Yuan, D.; Li, Y.; Ni, Z.; Pulay, P.; Li, W.; Li, S. Benchmark Relative Energies for Large Water Clusters with the Generalized Energy-Based Fragmentation Method.*J. Chem. Theory Comput.*2017,*13*, 2696– 2704, DOI: 10.1021/acs.jctc.7b00284Google Scholar75https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXntFeju7s%253D&md5=08bb06c6abcbed113677802f9752dc69Benchmark Relative Energies for Large Water Clusters with the Generalized Energy-Based Fragmentation MethodYuan, Dandan; Li, Yunzhi; Ni, Zhigang; Pulay, Peter; Li, Wei; Li, ShuhuaJournal of Chemical Theory and Computation (2017), 13 (6), 2696-2704CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The generalized energy-based fragmentation (GEBF) method has been applied to investigate relative energies of large water clusters (H2O)n (n = 32, 64) with the coupled-cluster singles and doubles with noniterative triple excitations (CCSD(T)) and second-order Moller-Plesset perturbation theory (MP2) at the complete basis set (CBS) limit. Here large water clusters are chosen to be representative structures sampled from mol. dynamics (MD) simulations of liq. water. Our calcns. show that the GEBF method is capable of providing highly accurate relative energies for these water clusters in a cost-effective way. We demonstrate that the relative energies from GEBF-MP2/CBS are in excellent agreement with those from GEBF-CCSD(T)/CBS for these water clusters. With the GEBF-CCSD(T)/CBS relative energies as the benchmark results, we have assessed the performance of several theor. methods widely used for ab initio MD simulations of liqs. and aq. solns. These methods include d. functional theory (DFT) with a no. of different functionals, MP2, and d. functional tight-binding (the third generation, DFTB3 in short). We find that MP2/aug-cc-pVDZ and several DFT methods (such as LC-ωPBE-D3 and ωB97XD) with the aug-cc-pVTZ basis set can provide satisfactory descriptions for these water clusters. Some widely used functionals (such as B3LYP, PBE0) and DFTB3 are not accurate enough for describing the relative energies of large water clusters. Although the basis set dependence of DFT is less than that of ab initio electron correlation methods, we recommend the combination of a few best functionals and large basis sets (at least aug-cc-pVTZ) in theor. studies on water clusters or aq. solns.**76**Li, W.; Ni, Z.; Li, S. Cluster-in-molecule local correlation method for post-Hartree-Fock calculations of large systems.*Mol. Phys.*2016,*114*, 1447, DOI: 10.1080/00268976.2016.1139755Google Scholar76https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhvFyhsrs%253D&md5=f4425783d8e50aac3d940428c1727f95Cluster-in-molecule local correlation method for post-Hartree-Fock calculations of large systemsLi, Wei; Ni, Zhigang; Li, ShuhuaMolecular Physics (2016), 114 (9), 1447-1460CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)Our recent developments on cluster-in-mol. (CIM) local correlation method are reviewed in this paper. In the CIM method, the correlation energy of a large system can be approx. obtained from electron correlation calcns. on a series of clusters, each of which contains a subset of occupied and virtual localised MOs in a certain region. The CIM method is a linear scaling method and its inherent parallelisation allows electron correlation calcns. of very large systems to be feasible at ordinary workstations. In the illustrative applications, this approach is applied to investigate the conformational energy differences, reaction barriers, and binding energies of large systems at the levels of Moller-Plesset perturbation theory and coupled-cluster theory.**77**Findlater, A. D.; Zahariev, F.; Gordon, M. S. Combined Fragment Molecular Orbital Cluster in Molecule Approach to Massively Parallel Electron Correlation Calculations for Large Systems.*J. Phys. Chem. A*2015,*119*, 3587, DOI: 10.1021/jp509266gGoogle Scholar77https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXkvVeltLw%253D&md5=a19f21db8ff082c4f4b77c7a96d9a41eCombined Fragment Molecular Orbital Cluster in Molecule Approach to Massively Parallel Electron Correlation Calculations for Large Systems.Findlater, Alexander D.; Zahariev, Federico; Gordon, Mark S.Journal of Physical Chemistry A (2015), 119 (15), 3587-3593CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)The local correlation "cluster-in-mol." (CIM) method is combined with the fragment MO (FMO) method, providing a flexible, massively parallel, and near-linear scaling approach to the calcn. of electron correlation energies for large mol. systems. Although the computational scaling of the CIM algorithm is already formally linear, previous knowledge of the Hartree-Fock (HF) ref. wave function and subsequent localized orbitals is required; therefore, extending the CIM method to arbitrarily large systems requires the aid of low-scaling/linear-scaling approaches to HF and orbital localization. Through fragmentation, the combined FMO-CIM method linearizes the scaling, with respect to system size, of the HF ref. and orbital localization calcns., achieving near-linear scaling at both the ref. and electron correlation levels. For the 20-residue alanine α helix, the preliminary implementation of the FMO-CIM method captures 99.6% of the MP2 correlation energy, requiring 21% of the MP2 wall time. The new method is also applied to solvated adamantine to illustrate the multilevel capability of the FMO-CIM method.**78**Stoll, H. Correlation energy of diamond.*Phys. Rev. B*1992,*46*, 6700, DOI: 10.1103/PhysRevB.46.6700Google Scholar78https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK38XlvFyrsrk%253D&md5=31fe8a45d647cc7d710b42bb49d41419Correlation energy of diamondStoll, HermannPhysical Review B: Condensed Matter and Materials Physics (1992), 46 (11), 6700-4CODEN: PRBMDO; ISSN:0163-1829.The correlation energy of diamond is detd. by means of increments obtained in ab initio calcns. for localized C-C bond orbitals and for pairs and triples of such bonds. The resulting correlation contribution to the cohesive energy is -0.129 a.u., which is approx. 85% of the exptl. value.**79**Li, W.; Li, S. Divide-and-conquer local correlation approach to the correlation energy of large molecules.*J. Chem. Phys.*2004,*121*, 6649, DOI: 10.1063/1.1792051Google Scholar79https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXotVKms7o%253D&md5=0679d39e6d4762edf30e3eb839a82149Divide-and-conquer local correlation approach to the correlation energy of large moleculesLi, Wei; Li, ShuhuaJournal of Chemical Physics (2004), 121 (14), 6649-6657CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A divide-and-conquer local correlation approach for correlation energy calcns. on large mols. is proposed for any post-Hartree-Fock correlation method. The main idea of this approach is to decomp. a large system into various fragments capped by their local environments. The total correlation energy of the whole system can be approx. obtained as the summation of correlation energies from all capped fragments, from which correlation energies from all adjacent caps are removed. This approach computationally achieves linear scaling even for medium-sized systems. Our test calcns. for a wide range of mols. using the 6-31G or 6-31G** basis set demonstrate that this simple approach recovers more than 99.0% of the conventional second-order Moller-Plesset perturbation theory and coupled cluster with single and double excitations correlation energies.**80**Li, W.; Piecuch, P.; Gour, J. R.; Li, S. Local correlation calculations using standard and renormalized coupled-cluster approaches.*J. Chem. Phys.*2009,*131*, 114109 DOI: 10.1063/1.3218842Google Scholar80https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXht1alsrjF&md5=b853f4b51c51a8675d2e0c0e8f0ea646Local correlation calculations using standard and renormalized coupled-cluster approachesLi, Wei; Piecuch, Piotr; Gour, Jeffrey R.; Li, ShuhuaJournal of Chemical Physics (2009), 131 (11), 114109/1-114109/30CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The linear scaling local correlation approach, termed "cluster-in-mol." (CIM), is extended to the coupled-cluster (CC) theory with singles and doubles (CCSD) and CC methods with singles, doubles, and noniterative triples, including CCSD(T) and the completely renormalized CR-CC(2,3) approach. The resulting CIM-CCSD, CIM-CCSD(T), and CIM-CR-CC(2,3) methods are characterized by (i) the linear scaling of the CPU time with the system size, (ii) the use of orthonormal orbitals in the CC subsystem calcns., (iii) the natural parallelism, (iv) the high computational efficiency, enabling calcns. for much larger systems and at higher levels of CC theory than previously possible, and (v) the purely noniterative character of local triples corrections. By comparing the results of the canonical and CIM-CC calcns. for normal alkanes and water clusters, it is shown that the CIM-CCSD, CIM-CCSD(T), and CIM-CR-CC(2,3) approaches accurately reproduce the corresponding canonical CC correlation and relative energies, while offering savings in the computer effort by orders of magnitude. (c) 2009 American Institute of Physics.**81**Fedorov, D. G.; Kitaura, K. Second order Møller-Plesset perturbation theory based upon the fragment molecular orbital method.*J. Chem. Phys.*2004,*121*, 2483, DOI: 10.1063/1.1769362Google Scholar81https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXlvFertLY%253D&md5=548708e044baadd2642840bc12412986Second order Moller-Plesset perturbation theory based upon the fragment molecular orbital methodFedorov, Dmitri G.; Kitaura, KazuoJournal of Chemical Physics (2004), 121 (6), 2483-2490CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The fragment MO (FMO) method was combined with the second order Moller-Plesset (MP2) perturbation theory. The accuracy of the method using the 6-31G* basis set was tested on (H2O)n, n = 16,32,64; α-helixes and β-strands of alanine n-mers, n = 10,20,40; as well as on (H2O)n, n = 16,32,64 using the 6-31++G** basis set. Relative to the regular MP2 results that could be afforded, the FMO2-MP2 error in the correlation energy did not exceed 0.003 a.u., the error in the correlation energy gradient did not exceed 0.000 05 a.u./bohr and the error in the correlation contribution to dipole moment did not exceed 0.03 debye. An approxn. reducing computational load based on fragment sepn. was introduced and tested. The FMO2-MP2 method demonstrated nearly linear scaling and drastically reduced the memory requirements of the regular MP2, making possible calcns. with several thousands basis functions using small Pentium clusters. As an example, (H2O)64 with the 6-31++G** basis set (1920 basis functions) can be run in 1 Gbyte RAM and it took 136 s on a 40-node Pentium4 cluster.**82**Guo, Y.; Li, W.; Li, S. Improved Cluster-in-Molecule Local Correlation Approach for Electron Correlation Calculation of Large Systems.*J. Phys. Chem. A*2014,*118*, 8996, DOI: 10.1021/jp501976xGoogle Scholar82https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhtVCru7nF&md5=72c632daa2fa3ee353383622564cfeecImproved Cluster-in-Molecule Local Correlation Approach for Electron Correlation Calculation of Large SystemsGuo, Yang; Li, Wei; Li, ShuhuaJournal of Physical Chemistry A (2014), 118 (39), 8996-9004CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)An improved cluster-in-mol. (CIM) local correlation approach is developed to allow electron correlation calcns. of large systems more accurate and faster. We have proposed a refined strategy of constructing virtual LMOs of various clusters, which is suitable for basis sets of various types. To recover medium-range electron correlation, which is important for quant. descriptions of large systems, we find that a larger distance threshold (ξ) is necessary for highly accurate results. Our illustrative calcns. show that the present CIM-MP2 (second-order Moller-Plesset perturbation theory, MP2) or CIM-CCSD (coupled cluster singles and doubles, CCSD) scheme with a suitable ξ value is capable of recovering more than 99.8% correlation energies for a wide range of systems at different basis sets. The present CIM-MP2 scheme can provide reliable relative energy differences as the conventional MP2 method for secondary structures of polypeptides.**83**Kobayashi, M.; Imamura, Y.; Nakai, H. Alternative linear-scaling methodology for the second-order Møller-Plesset perturbation calculation based on the divide-and-conquer method.*J. Chem. Phys.*2007,*127*, 074103 DOI: 10.1063/1.2761878Google Scholar83https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXpsFGntLc%253D&md5=1207a5b4335f7cd457ad57343bd93e3fAlternative linear-scaling methodology for the second-order Moeller-Plesset perturbation calculation based on the divide-and-conquer methodKobayashi, Masato; Imamura, Yutaka; Nakai, HiromiJournal of Chemical Physics (2007), 127 (7), 074103/1-074103/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A new scheme for obtaining the approx. correlation energy in the divide-and-conquer (DC) method of Yang [Phys. Rev. Lett. 66, 1438 (1991)] is presented. In this method, the correlation energy of the total system is evaluated by summing up subsystem contributions, which are calcd. from subsystem orbitals based on a scheme for partitioning the correlation energy. We applied this method to the second-order Moeller-Plesset perturbation theory (MP2), which we call DC-MP2. Numerical assessment revealed that this scheme provides a reliable correlation energy with significantly less computational cost than the conventional MP2 calcn.**84**Ziółkowski, M.; Jansík, B.; Kjærgaard, T.; Jørgensen, P. Linear scaling coupled cluster method with correlation energy based error control.*J. Chem. Phys.*2010,*133*, 014107 DOI: 10.1063/1.3456535Google Scholar84https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXosV2qsL8%253D&md5=75346bd355c27ddb06a6136a92fdb894Linear scaling coupled cluster method with correlation energy based error controlZiolkowski, Marcin; Jansik, Branislav; Kjaergaard, Thomas; Jorgensen, PoulJournal of Chemical Physics (2010), 133 (1), 014107/1-014107/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Coupled cluster calcns. can be carried out for large mol. systems via a set of calcns. that use small orbital fragments of the full MO space. The error in the correlation energy of the full mol. system is controlled by the precision in the small fragment calcns. The detn. of the orbital spaces for the small orbital fragments is black box in the sense that it does not depend on any user-provided mol. fragmentation, rather orbital spaces are carefully selected and extended during the calcn. to give fragment energies of a specified precision. The computational method scales linearly with the size of the mol. system and is massively parallel. (c) 2010 American Institute of Physics.**85**Kjærgaard, T. The Laplace transformed divide-expand-consolidate resolution of the identity second-order Møller-Plesset perturbation (DEC-LT-RIMP2) theory method.*J. Chem. Phys.*2017,*146*, 044103 DOI: 10.1063/1.4973710Google Scholar85https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhs1Shsr8%253D&md5=5962de7379c72f2d6af0c4c36b625047The Laplace transformed divide-expand-consolidate resolution of the identity second-order Moller-Plesset perturbation (DEC-LT-RIMP2) theory methodKjaergaard, ThomasJournal of Chemical Physics (2017), 146 (4), 044103/1-044103/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The divide-expand-consolidate resoln. of the identity second-order Moller-Plesset perturbation (DEC-RI-MP2) theory method introduced in Baudin et al. [J. Chem. Phys. 144, 054102 (2016)] is significantly improved by introducing the Laplace transform of the orbital energy denominator in order to construct the double amplitudes directly in the local basis. Furthermore, this paper introduces the auxiliary redn. procedure, which reduces the set of the auxiliary functions employed in the individual fragments. The resulting Laplace transformed divide-expand-consolidate resoln. of the identity second-order Moller-Plesset perturbation method is applied to the insulin mol. where we obtain a factor 9.5 speedup compared to the DEC-RI-MP2 method. (c) 2017 American Institute of Physics.**86**Anacker, T.; Tew, D. P.; Friedrich, J. First UHF Implementation of the Incremental Scheme for Open-Shell Systems.*J. Chem. Theory Comput.*2016,*12*, 65, DOI: 10.1021/acs.jctc.5b00933Google Scholar86https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhslKlsbnE&md5=39dfae953b1302f905594f1b6be9b198First UHF Implementation of the Incremental Scheme for Open-Shell SystemsAnacker, Tony; Tew, David P.; Friedrich, JoachimJournal of Chemical Theory and Computation (2016), 12 (1), 65-78CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The incremental scheme makes it possible to compute CCSD(T) correlation energies to high accuracy for large systems. We present the first extension of this fully automated black-box approach to open-shell systems using an UHF (UHF) wave function, extending the efficient domain-specific basis set approach to handle open-shell refs. We test our approach on a set of org. and metal org. structures and mol. clusters and demonstrate std. deviations from canonical CCSD(T) values of only 1.35 kJ/mol using a triple ζ basis set. We find that the incremental scheme is significantly more cost-effective than the canonical implementation even for relatively small systems and that the ease of parallelization makes it possible to perform high-level calcns. on large systems in a few hours on inexpensive computers. We show that the approxns. that make our approach widely applicable are significantly smaller than both the basis set incompleteness error and the intrinsic error of the CCSD(T) method, and we further demonstrate that incremental energies can be reliably used in extrapolation schemes to obtain near complete basis set limit CCSD(T) reaction energies for large systems.**87**Zhang, J.; Dolg, M. Third-Order Incremental Dual-Basis Set Zero-Buffer Approach for Large High-Spin Open-Shell Systems.*J. Chem. Theory Comput.*2015,*11*, 962, DOI: 10.1021/ct501052eGoogle Scholar87https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhsF2msLY%253D&md5=94659836d62fdc83da583c1aa5c50d7eThird-Order Incremental Dual-Basis Set Zero-Buffer Approach for Large High-Spin Open-Shell SystemsZhang, Jun; Dolg, MichaelJournal of Chemical Theory and Computation (2015), 11 (3), 962-968CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The third-order incremental dual-basis set zero-buffer approach (inc3-db-B0) is an efficient, accurate, and black-box quantum chem. method for obtaining correlation energies of large systems, and it has been successfully applied to many real chem. problems. In this work, we extend this approach to high-spin open-shell systems. In the open-shell approach, we will first decomp. the occupied orbitals of a system into several domains by a K-means clustering algorithm. The essential part is that we preserve the active (singly occupied) orbitals in all the calcns. of the domain correlation energies. The duplicated contributions of the active orbitals to the correlation energy are subtracted from the incremental expansion. All techniques of truncating the virtual space such as the B0 approxn. can be applied. This open-shell inc3-db-B0 approach is combined with the CCSD and CCSD(T) methods and applied to the computations of a singlet-triplet gap and an electron detachment process. Our approach exhibits an accuracy better than 0.6 kcal/mol or 0.3 eV compared with the std. implementation, while it saves a large amt. of the computational time and can be efficiently parallelized.**88**Kállay, M. Linear-scaling implementation of the direct random-phase approximation.*J. Chem. Phys.*2015,*142*, 204105 DOI: 10.1063/1.4921542Google Scholar88https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXpt1Sjsb4%253D&md5=a66b1691368dd94596a8052c569de4d2Linear-scaling implementation of the direct random-phase approximationKallay, MihalyJournal of Chemical Physics (2015), 142 (20), 204105/1-204105/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We report the linear-scaling implementation of the direct RPA (dRPA) for closed-shell mol. systems. As a bonus, linear-scaling algorithms are also presented for the second-order screened exchange extension of dRPA as well as for the second-order Moller-Plesset (MP2) method and its spin-scaled variants. Our approach is based on an incremental scheme which is an extension of our previous local correlation method [Rolik et al., J. Chem. Phys. 139, 094105 (2013)]. The approach extensively uses local natural orbitals to reduce the size of the MO basis of local correlation domains. In addn., we also demonstrate that using natural auxiliary functions [M. Kallay, J. Chem. Phys. 141, 244113 (2014)], the size of the auxiliary basis of the domains and thus that of the three-center Coulomb integral lists can be reduced by an order of magnitude, which results in significant savings in computation time. The new approach is validated by extensive test calcns. for energies and energy differences. Our benchmark calcns. also demonstrate that the new method enables dRPA calcns. for mols. with more than 1000 atoms and 10 000 basis functions on a single processor. (c) 2015 American Institute of Physics.**89**Nagy, P. R.; Samu, G.; Kállay, M. Optimization of the linear-scaling local natural orbital CCSD(T) method: Improved algorithm and benchmark applications.*J. Chem. Theory Comput.*2018,*14*, 4193, DOI: 10.1021/acs.jctc.8b00442Google Scholar89https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXht1KgurnJ&md5=b241b1582220a712becf5185bcf62cbcOptimization of the Linear-Scaling Local Natural Orbital CCSD(T) Method: Improved Algorithm and Benchmark ApplicationsNagy, Peter R.; Samu, Gyula; Kallay, MihalyJournal of Chemical Theory and Computation (2018), 14 (8), 4193-4215CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)An optimized implementation of the local natural orbital (LNO) coupled-cluster (CC) with single-, double-, and perturbative triple excitations [LNO-CCSD(T)] method is presented. The integral-direct, in-core, highly efficient domain construction technique of the authors' local second-order Moller-Plesset (LMP2) scheme is extended to the CC level. The resulting scheme, which is also suitable for general-order LNO-CC calcns., inherits the beneficial properties of the LMP2 approach, such as the asymptotically linear-scaling operation count, the asymptotically const. data storage requirement, and the completely independent domain calcns. In addn. to integrating the authors' recent redundancy-free LMP2 and Laplace-transformed (T) algorithms with the LNO-CCSD(T) code, the memory demand, the domain and LNO construction, the auxiliary basis compression, and the previously rate-detg. two-external integral transformation have been significantly improved. The accuracy of all of the approxns. is carefully examd. on medium-sized to large systems to det. reasonably tight default truncation thresholds. The authors' benchmark calcns., performed on mols. of up to 63 atoms, show that the optimized method with the default settings provides av. correlation and reaction energy errors of <0.07% and 0.34 kcal/mol, resp., compared to the canonical CCSD(T) ref. The efficiency of the present LNO-CCSD(T) implementation is demonstrated on realistic, three-dimensional examples. Using the new code, an LNO-CCSD(T) correlation energy calcn. with a triple-ζ basis set is feasible on a single processor for a protein mol. including 2380 atoms and >44000 AOs.**90**Nagy, P. R.; Kállay, M. Approaching the basis set limit of CCSD(T) energies for large molecules with local natural orbital coupled-cluster methods.*J. Chem. Theory Comput.*2019,*15*, 5275, DOI: 10.1021/acs.jctc.9b00511Google Scholar90https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhs12jtrrM&md5=c4b7d13c965f077aad98b9c869a098baApproaching the Basis Set Limit of CCSD(T) Energies for Large Molecules with Local Natural Orbital Coupled-Cluster MethodsNagy, Peter R.; Kallay, MihalyJournal of Chemical Theory and Computation (2019), 15 (10), 5275-5298CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Recent optimization efforts and extensive benchmark applications are presented illustrating the accuracy and efficiency of the linear-scaling local natural orbital (LNO) coupled-cluster single-, double-, and perturbative triple-excitations [CCSD(T)] method. A composite threshold combination hierarchy (Loose, Normal, Tight, etc.) is introduced, which enables black box convergence tests and is useful to est. the accuracy of the LNO-CCSD(T) energies with respect to CCSD(T). We also demonstrate that the complete basis set limit (CBS) of LNO-CCSD(T) energies can be reliably approached via basis set extrapolation using large basis sets including diffuse functions. Where ref. CCSD(T) results are available, the mean (max.) abs. errors of the LNO-CCSD(T) reaction and intermol. interaction energies with the default Normal threshold combination are below 0.2-0.3 (0.6-1.0) kcal/mol, while the same measures with the Tight setting are 0.1 (0.2-0.5) kcal/mol for all the tested systems including highly complicated cases. The performance of LNO-CCSD(T) is also compared with that of other popular local CCSD(T) schemes. The exceptionally low hardware requirements of the present scheme enables the routine calcn. of benchmark-quality energy differences within chem. accuracy of CCSD(T)/CBS for systems including a few hundred atoms. LNO-CCSD(T)/CBS calcns. can also be performed for more than 1000 atoms with 45,000 AOs using a single, six-core CPU, about 100 GB memory, and comparable disk space.**91**Koch, H.; Sánchez de Merás, A. M. Size-intensive decomposition of orbital energy denominators.*J. Chem. Phys.*2000,*113*, 508, DOI: 10.1063/1.481910Google Scholar91https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3cXksV2iu78%253D&md5=5247869528076abfe4720efc357b4018Size-intensive decomposition of orbital energy denominatorsKoch, Henrik; Sanchez de Meras, AlfredoJournal of Chemical Physics (2000), 113 (2), 508-513CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We introduce an alternative to Almlof and Haser's Laplace transform decompn. of orbital energy denominators used in obtaining reduced scaling algorithms in perturbation theory based methods. The new decompn. is based on the Cholesky decompn. of pos. semidefinite matrixes. Orbital denominators have a particular short and size-intensive Cholesky decompn. The main advantage in using the Cholesky decompn., besides the shorter expansion, is the systematic improvement of the results without the penalties encountered in the Laplace transform decompn. when changing the no. of integration points in order to control the convergence. Applications will focus on the coupled-cluster singles and doubles model including connected triples corrections [CCSD(T)], and several numerical examples are discussed.**92**Rolik, Z.; Szegedy, L.; Ladjánszki, I.; Ladóczki, B.; Kállay, M. An efficient linear-scaling CCSD(T) method based on local natural orbitals.*J. Chem. Phys.*2013,*139*, 094105 DOI: 10.1063/1.4819401Google Scholar92https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXhtl2rurrL&md5=4c42884bdc3e611e66f8642ba04f7847An efficient linear-scaling CCSD(T) method based on local natural orbitalsRolik, Zoltan; Szegedy, Lorant; Ladjanszki, Istvan; Ladoczki, Bence; Kallay, MihalyJournal of Chemical Physics (2013), 139 (9), 094105/1-094105/17CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)An improved version of our general-order local coupled-cluster (CC) approach and its efficient implementation at the CC singles and doubles with perturbative triples CCSD(T) level is presented. The method combines the cluster-in-mol. approach of with frozen natural orbital (NO) techniques. To break down the unfavorable fifth-power scaling of our original approach a two-level domain construction algorithm has been developed. First, an extended domain of localized MOs (LMOs) is assembled based on the spatial distance of the orbitals. The necessary integrals are evaluated and transformed in these domains invoking the d. fitting approxn. In the second step, for each occupied LMO of the extended domain a local subspace of occupied and virtual orbitals is constructed including approx. second-order Moller-Plesset NOs. The CC equations are solved and the perturbative corrections are calcd. in the local subspace for each occupied LMO using a highly-efficient CCSD(T) code, which was optimized for the typical sizes of the local subspaces. The total correlation energy is evaluated as the sum of the individual contributions. The computation time of our approach scales linearly with the system size, while its memory and disk space requirements are independent thereof. Test calcns. demonstrate that currently our method is one of the most efficient local CCSD(T) approaches and can be routinely applied to mols. of up to 100 atoms with reasonable basis sets. (c) 2013 American Institute of Physics.**93**Rolik, Z.; Kállay, M. A general-order local coupled-cluster method based on the cluster-in-molecule approach.*J. Chem. Phys.*2011,*135*, 104111 DOI: 10.1063/1.3632085Google Scholar93https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhtFGnur%252FM&md5=e6d596fc29ff34ef150cb6a119874765A general-order local coupled-cluster method based on the cluster-in-molecule approachRolik, Zoltan; Kallay, MihalyJournal of Chemical Physics (2011), 135 (10), 104111/1-104111/18CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A general-order local coupled-cluster (CC) method is presented which has the potential to provide accurate correlation energies for extended systems. Our method combines the cluster-in-mol. approach of with the frozen natural orbital (NO) techniques widely used for the cost redn. of correlation methods. The occupied MOs are localized, and for each occupied MO a local subspace of occupied and virtual orbitals is constructed using approx. Moller-Plesset NOs. The CC equations are solved and the correlation energies are calcd. in the local subspace for each occupied MO, while the total correlation energy is evaluated as the sum of the individual contributions. The size of the local subspaces and the accuracy of the results can be controlled by varying only one parameter, the threshold for the occupation no. of NOs which are included in the subspaces. Though our local CC method in its present form scales as the fifth power of the system size, our benchmark calcns. show that it is still competitive for the CC singles and doubles (CCSD) and the CCSD with perturbative triples CCSD(T) approaches. For higher order CC methods, the redn. in computation time is more pronounced, and the new method enables calcns. for considerably bigger mols. than before with a reasonable loss in accuracy. We also demonstrate that the independent calcn. of the correlation contributions allows for a higher order description of the chem. more important segments of the mol. and a lower level treatment of the rest delivering further significant savings in computer time. (c) 2011 American Institute of Physics.**94**Ma, Q.; Werner, H.-J. Scalable Electron Correlation Methods. 7. Local Open-Shell Coupled-Cluster Methods Using Pair Natural Orbitals: PNO-RCCSD and PNO-UCCSD.*J. Chem. Theory Comput.*2020,*16*, 3135, DOI: 10.1021/acs.jctc.0c00192Google Scholar94https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXmvV2qur4%253D&md5=f2ba7c898f14c392c4f12dff63889807Scalable Electron Correlation Methods. 7. Local Open-Shell Coupled-Cluster Methods Using Pair Natural Orbitals: PNO-RCCSD and PNO-UCCSDMa, Qianli; Werner, Hans-JoachimJournal of Chemical Theory and Computation (2020), 16 (5), 3135-3151CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present well-parallelized local implementations of high-spin open-shell coupled cluster methods with single and double excitations (CCSD) using pair natural orbitals (PNOs). The methods are based on the spin-orbital coupled cluster theory using restricted open-shell Hartree-Fock (ROHF) ref. functions. Two variants, namely, PNO-UCCSD and PNO-RCCSD are implemented and compared. In PNO-UCCSD, the coupled cluster amplitudes are spin-unrestricted, while in PNO-RCCSD the linear terms are spin-adapted by a spin-projection approach as described in. Near linear scaling of the computational cost with the no. of correlated electrons is achieved by applying domain and pair approxns. The PNOs are spin-independent and obtained using a semicanonical spin-restricted MP2 approxn. with large domains of projected AOs (PAOs). The pair approxns. of our previously described closed-shell PNO-LCCSD method are carefully revised so that they are compatible to the UCCSD theory, and PNO-UCCSD or PNO-RCCSD calcns. for closed-shell mols. yield exactly the same results as corresponding spin-free closed-shell PNO-LCCSD calcns. The convergence of the results with respect to the thresholds and options that control the domain and pair approxns. is demonstrated. It is found that large domains are required for the single excitations in open-shell calcns. in order to obtain converged results. In general, the errors of relative energies caused by the local approxns. can be reduced to below 1 kcal mol-1, even for difficult cases. Presently, PNO-RCCSD and PNO-UCCSD calcns. for mols. with 100-200 atoms and augmented triple-ζ basis sets can be carried out in a few hours of elapsed time using ∼ 100 CPU cores. In addn., the program is also capable of performing distinguishable cluster (PNO-RDCSD and PNO-UDCSC) calcns. The present work is a crit. step in developing fully local open-shell PNO-RCCSD(T)-F12 methods.**95**Ma, Q.; Werner, H.-J. Scalable Electron Correlation Methods. 8. Explicitly Correlated Open-Shell Coupled-Cluster with Pair Natural Orbitals PNO-RCCSD(T)-F12 and PNO-UCCSD(T)-F12.*J. Chem. Theory Comput.*2021,*17*, 902, DOI: 10.1021/acs.jctc.0c01129Google Scholar95https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXjslOrtw%253D%253D&md5=494aac943d70b4863645029e3ae157e4Scalable Electron Correlation Methods. 8. Explicitly Correlated Open-Shell Coupled-Cluster with Pair Natural Orbitals PNO-RCCSD(T)-F12 and PNO-UCCSD(T)-F12Ma, Qianli; Werner, Hans-JoachimJournal of Chemical Theory and Computation (2021), 17 (2), 902-926CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present explicitly correlated open-shell pair natural orbital local coupled-cluster methods, PNO-RCCSD(T)-F12 and PNO-UCCSD(T)-F12. The methods are extensions of our previously reported PNO-R/UCCSD methods (J. Chem. Theory Comput., 2020, 16, 3135-3151, https://pubs.acs.org/doi/10.1021/acs.jctc.0c00192) with addns. of explicit correlation and perturbative triples corrections. The explicit correlation treatment follows the spin-orbital CCSD-F12b theory using Ansatz 3*A, which is found to yield comparable or better basis set convergence than the more rigorous Ansatz 3C in computed ionization potentials and reaction energies using double- to quaduple-ζ basis sets. The perturbative triples correction is adapted from the spin-orbital (T) theory to use triples natural orbitals (TNOs). To address the coupling due to off-diagonal Fock matrix elements, the local triples amplitudes are iteratively solved using small domains of TNOs, and a semicanonical (T0) domain correction with larger domains is applied to reduce the domain errors. The performance of the methods is demonstrated through benchmark calcns. on ionization potentials, radical stabilization energies, reaction energies of fragmentations and rearrangements in radical cations, and spin-state energy differences of iron complexes. For a few test sets where canonical calcns. are feasible, PNO-RCCSD(T)-F12 results agree with the canonical ones to within 0.4 kcal mol-1, and this max. error is reduced to below 0.2 kcal mol-1 when large local domains are used. For larger systems, results using different thresholds for the local approxns. are compared to demonstrate that 1 kcal mol-1 level of accuracy can be achieved using our default settings. For a couple of difficult cases, it is demonstrated that the errors from individual approxns. are only a fraction of 1 kcal mol-1, and the overall accuracy of the method does not rely on error compensations. In contrast to canonical calcns., the use of spin-orbitals does not lead to a significant increase of computational time and memory usage in the most expensive steps of PNO-R/UCCSD(T)-F12 calcns. The only exception is the iterative soln. of the (T) amplitudes, which can be avoided without significant errors by using a perturbative treatment of the off-diagonal coupling, known as (T1) approxn. For most systems, even the semicanonical approxn. (T0) leads only to small errors in relative energies. Our program is well parallelized and capable of computing accurate correlation energies for mols. with 100-200 atoms using augmented triple-ζ basis sets in less than a day of elapsed time on a small computer cluster.**96**Hansen, A.; Liakos, D. G.; Neese, F. Efficient and accurate local single reference correlation methods for high-spin open-shell molecules using pair natural orbitals.*J. Chem. Phys.*2011,*135*, 214102 DOI: 10.1063/1.3663855Google Scholar96https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhsFOitbzJ&md5=eca169c948ade43adb095c23be065b7dEfficient and accurate local single reference correlation methods for high-spin open-shell molecules using pair natural orbitalsHansen, Andreas; Liakos, Dimitrios G.; Neese, FrankJournal of Chemical Physics (2011), 135 (21), 214102/1-214102/20CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A prodn. level implementation of the high-spin open-shell (spin unrestricted) single ref. coupled pair, quadratic CI and coupled cluster methods with up to doubly excited determinants in the framework of the local pair natural orbital (LPNO) concept is reported. This work is an extension of the closed-shell LPNO methods developed earlier. The internal space is spanned by localized orbitals, while the external space for each electron pair is represented by a truncated PNO expansion. The laborious integral transformation assocd. with the large no. of PNOs becomes feasible through the extensive use of d. fitting (resoln. of the identity (RI)) techniques. Tech. complications arising for the open-shell case and the use of quasi-restricted orbitals for the construction of the ref. determinant are discussed in detail. As in the closed-shell case, only three cutoff parameters control the av. no. of PNOs per electron pair, the size of the significant pair list, and the no. of contributing auxiliary basis functions per PNO. The chosen threshold default values ensure robustness and the results of the parent canonical methods are reproduced to high accuracy. Comprehensive numerical tests on abs. and relative energies as well as timings consistently show that the outstanding performance of the LPNO methods carries over to the open-shell case with minor modifications. Finally, hyperfine couplings calcd. with the variational LPNO-CEPA/1 method, for which a well-defined expectation value type d. exists, indicate the great potential of the LPNO approach for the efficient calcn. of mol. properties. (c) 2011 American Institute of Physics.**97**Guo, Y.; Riplinger, C.; Liakos, D. G.; Becker, U.; Saitow, M.; Neese, F. Linear scaling perturbative triples correction approximations for open-shell domain-based local pair natural orbital coupled cluster singles and doubles theory [DLPNO-CCSD(T0/T)].*J. Chem. Phys.*2020,*152*, 024116 DOI: 10.1063/1.5127550Google Scholar97https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXpt1Kgsg%253D%253D&md5=5f26aa0adab02b69aee1e4f3a349c50fLinear scaling perturbative triples correction approximations for open-shell domain-based local pair natural orbital coupled cluster singles and doubles theory [DLPNO-CCSD(T0/T)]Guo, Yang; Riplinger, Christoph; Liakos, Dimitrios G.; Becker, Ute; Saitow, Masaaki; Neese, FrankJournal of Chemical Physics (2020), 152 (2), 024116CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The coupled cluster method with single-, double-, and perturbative triple excitations [CCSD(T)] is considered to be one of the most reliable quantum chem. theories. However, the steep scaling of CCSD(T) has limited its application to small or medium-sized systems for a long time. In our previous work, the linear scaling domain based local pair natural orbital CCSD variant (DLPNO-CCSD) has been developed for closed-shell and open-shell. However, it is known from extensive benchmark studies that triple-excitation contributions are important to reach chem. accuracy. In the present work, two linear scaling (T) approxns. for open-shell DLPNO-CCSD are implemented and compared: (a) an algorithm based on the semicanonical approxn., in which off-diagonal Fock matrix elements in the occupied space are neglected [referred to as DLPNO-(T0)]; and (b) an improved algorithm in which the triples amplitudes are computed iteratively [referred to as DLPNO-(T)]. This work is based on the previous open-shell DLPNO-CCSD algorithm [M. Saitow et al., J. Chem. Phys. 146, 164105 (2017)] as well as the iterative (T) correction for closed-shell systems [Y. Guo et al., J. Chem. Phys. 148, 011101 (2018)]. Our results show that the new open-shell perturbative corrections, DLPNO-(T0/T), can predict accurate abs. and relative correlation energies relative to the canonical ref. calcns. with the same basis set. The abs. energies from DLPNO-(T) are significantly more accurate than those of DLPNO-(T0). The addnl. computational effort of DLPNO-(T) relative to DLPNO-(T0) is a factor of 4 on av. We report calcns. on systems with more than 4000 basis functions. (c) 2020 American Institute of Physics.**98**Kumar, A.; Neese, F.; Valeev, E. F. Explicitly correlated coupled cluster method for accurate treatment of open-shell molecules with hundreds of atoms.*J. Chem. Phys.*2020,*153*, 094105 DOI: 10.1063/5.0012753Google Scholar98https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhsl2isbnM&md5=d631f824055ab80dbc8742390caa824eExplicitly correlated coupled cluster method for accurate treatment of open-shell molecules with hundreds of atomsKumar, Ashutosh; Neese, Frank; Valeev, Edward F.Journal of Chemical Physics (2020), 153 (9), 094105CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present a near-linear scaling formulation of the explicitly correlated coupled-cluster singles and doubles with the perturbative triples method [CCSD(T)F12] for high-spin states of open-shell species. The approach is based on the conventional open-shell CCSD formalism [M. Saitow et al., J. Chem. Phys. 146, 164105 (2017)] utilizing the domain local pair-natural orbitals (DLPNO) framework. The use of spin-independent set of pair-natural orbitals ensures exact agreement with the closed-shell formalism reported previously, with only marginally impact on the cost (e.g., the open-shell formalism is only 1.5 times slower than the closed-shell counterpart for the C160H322 n-alkane, with the measured size complexity of ≈ 1.2). Evaluation of coupled-cluster energies near the complete-basis-set (CBS) limit for open-shell systems with more than 550 atoms and 5000 basis functions is feasible on a single multi-core computer in less than 3 days. The aug-cc-pVTZ DLPNO-CCSD(T)F12 contribution to the heat of formation for the 50 largest mols. among the 348 core combustion species benchmark set [J. Klippenstein et al., J. Phys. Chem. A 121, 6580-6602 (2017)] had root-mean-square deviation (RMSD) from the extrapolated CBS CCSD(T) ref. values of 0.3 kcal/mol. For a more challenging set of 50 reactions involving small closed- and open-shell mols. [G. Knizia et al., J. Chem. Phys. 130, 054104 (2009)], the aug-cc-pVQ( + d)Z DLPNO-CCSD(T)F12 yielded a RMSD of ∼0.4 kcal/mol with respect to the CBS CCSD(T) est. (c) 2020 American Institute of Physics.**99**Angeli, C.; Cimiraglia, R.; Evangelisti, S.; Leininger, T.; Malrieu, J.-P. Introduction of n-electron valence states for multireference perturbation theory.*J. Chem. Phys.*2001,*114*, 10252, DOI: 10.1063/1.1361246Google Scholar99https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3MXkt1antro%253D&md5=1bd85c0ec505be43e660bfe9820ab455Introduction of n-electron valence states for multireference perturbation theoryAngeli, C.; Cimiraglia, R.; Evangelisti, S.; Leininger, T.; Malrieu, J.-P.Journal of Chemical Physics (2001), 114 (23), 10252-10264CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The present work presents three second-order perturbative developments from a complete active space (CAS) zero-order wave function, which are strictly additive with respect to mol. dissocn. and intruder state free. They differ by the degree of contraction of the outer-space perturbers. Two types of zero-order Hamiltonians are proposed, both are bielectronic, incorporating the interactions between electrons in the active orbitals, therefore introducing a rational balance between the zero-order wave function and the outer-space. The use of Dyall's Hamiltonian, which puts the active electrons in a fixed core field, and of a partially contracted formalism seems a promising compromise. The formalism is generalizable to multireference spaces which are parts of a CAS. A few test applications of the simplest variant developed in this paper illustrate its potentialities.**100**Lauderdale, W. J.; Stanton, J. F.; Gauss, J.; Watts, J. D.; Bartlett, R. J. Many-body perturbation theory with a restricted open-shell Hartree-Fock reference.*Chem. Phys. Lett.*1991,*187*, 21, DOI: 10.1016/0009-2614(91)90478-RGoogle Scholar100https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK38XisVajt7w%253D&md5=992fc72865b5b8d04b09bcbbf6b05f85Many-body perturbation theory with a restricted open-shell Hartree-Fock referenceLauderdale, Walter J.; Stanton, John F.; Gauss, Jurgen; Watts, John D.; Bartlett, Rodney J.Chemical Physics Letters (1991), 187 (1-2), 21-8CODEN: CHPLBC; ISSN:0009-2614.A new, efficient ROHF-based MBPT method is presented. The method, which is noniterative, invariant to transformations among occupied or virtual orbitals, and generalizable to any order, is illustrated by application to the UHF spin-contaminated CN radical and the H + OCH2 transition state.**101**Knowles, P. J.; Andrews, J. S.; Amos, R. D.; Handy, N. C.; Pople, J. A. Restricted Møller-Plesset theory for open-shell molecules.*Chem. Phys. Lett.*1991,*186*, 130, DOI: 10.1016/S0009-2614(91)85118-GGoogle Scholar101https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK38XivVentA%253D%253D&md5=1f8bf30832c410b21f4330f6e9de39d4Restricted Moeller-Plesset theory for open-shell moleculesKnowles, Peter J.; Andrews, Jamie S.; Amos, Roger D.; Handy, Nicholas C.; Pople, John A.Chemical Physics Letters (1991), 186 (2-3), 130-6CODEN: CHPLBC; ISSN:0009-2614.Moeller-Plesset perturbation-theory calcns. are examd. for open-shell mols. based on a spin-RHF ref. wavefunction through the development of a new prescription for obtaining unique semi-canonical orbitals. These orbitals, which are different for α and β spins while avoiding the spin contamination present in UHF ref. functions, satisfy the criteria on which Koopmans's theorem is based, lending justification to their use in defining a zeroth-order Hamiltonian for perturbation theory. This new and straightforward Moeller-Plesset theory for open-shell mols. is called RMP theory. The convergence of the RMP series is examd. to high order, and shows the greatly improved convergence characteristics also found with the authors' alternative ROMP theory. For the mols. NH2(re, 1.5re, 2re) and CN, RMP2 energies are substantially lower than UMP2 energies.**102**Neese, F. Importance of Direct Spin-Spin Coupling and Spin-Flip Excitations for the Zero-Field Splittings of Transition Metal Complexes: A Case Study.*J. Am. Chem. Soc.*2006,*128*, 10213, DOI: 10.1021/ja061798aGoogle Scholar102https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28Xms1Sks7w%253D&md5=b6358991e999276be8095c58b9530c78Importance of Direct Spin-Spin Coupling and Spin-Flip Excitations for the Zero-Field Splittings of Transition Metal Complexes: A Case StudyNeese, FrankJournal of the American Chemical Society (2006), 128 (31), 10213-10222CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)This work reports the evaluation of several theor. approaches to the zero-field splitting (ZFS) in transition metal complexes. The exptl. well-known complex [Mn(acac)3] is taken as an example. The direct spin-spin contributions to the ZFS have been calcd. on the basis of d. functional theory (DFT) or complete active space SCF (CASSCF) wave functions and have been found to be much more important than previously assumed. The contributions of the direct term may exceed ∼1 cm-1 in magnitude and therefore cannot be neglected in any treatment that aims at a realistic quant. modeling of the ZFS. In the DFT framework, two different variants to treat the spin-orbit coupling (SOC) term have been evaluated. The first approach is based on previous work by Pederson, Khanna, and Kortus, and the second is based on a "quasi-restricted" DFT treatment which is rooted in our previous work on ZFS. Both approaches provide very similar results and underestimate the SOC contribution to the ZFS by a factor of 2 or more. The SOC is represented by an accurate multicenter spin-orbit mean-field (SOMF) approxn. which is compared to the popular effective DFT potential-derived SOC operator. In addn. to the DFT results, direct "infinite order" ab initio calcns. of the SOC contribution to the ZFS based on CASSCF wave functions, the spectroscopy-oriented CI (SORCI), and the difference-dedicated CI (DDCI) approach are reported. In general, the multireference ab initio results provide a more realistic description of the ZFS in [Mn(acac)3]. The conclusions likely carry over to many other systems. This is attributed to the explicit treatment of the multiplet effects which are of dominant importance, since the calcns. demonstrate that, even in the high-spin d4 system Mn(III), the spin-flip excitations make the largest contribution to the SOC. It is demonstrated that the ab initio methods can be used even for somewhat larger mols. (the present calcns. were done with more than 500 basis functions) in a reasonable time frame. Much more economical but still fairly reasonable results have been achieved with the INDO/S treatment based on CASSCF and SOC-CI wave functions.**103**Hégely, B.; Nagy, P. R.; Kállay, M. Dual basis set approach for density functional and wave function embedding schemes.*J. Chem. Theory Comput.*2018,*14*, 4600, DOI: 10.1021/acs.jctc.8b00350Google Scholar103https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXhtl2js7vF&md5=993851ee9db874e16cd6868bc37164deDual Basis Set Approach for Density Functional and Wave Function Embedding SchemesHegely, Bence; Nagy, Peter R.; Kallay, MihalyJournal of Chemical Theory and Computation (2018), 14 (9), 4600-4615CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A dual basis (DB) approach is proposed which is suitable for the redn. of the computational expenses of the Hartree-Fock, Kohn-Sham, and wave function-based correlation methods. The approach is closely related to the DB approxn. of Head-Gordon and co-workers [ J. Chem. Phys. 2006, 125, 074108] but specifically designed for embedding calcns. The new approach is applied to our variant of the projector-based embedding theory utilizing the Huzinaga-equation, multilevel local correlation methods, and combined d. functional-multilevel local correlation approxns. The performance of the resulting DB d. functional and wave function embedding methods is evaluated in extensive benchmark calcns. and also compared to that of the corresponding embedding schemes exploiting the mixed-basis approxn. Our results show that, with an appropriate combination of basis sets, the DB approach significantly speeds up the embedding calcns., and, for chem. processes where the electronic structure considerably changes, it is clearly superior to the mixed-basis approxn. The results also demonstrate that the DB approach, if integrated with the mixed-basis approxn., efficiently eliminates the major weakness of the latter, and the combination of the DB and mixed-basis schemes is the most efficient strategy to accelerate embedding calcns.**104**Hégely, B.; Nagy, P. R.; Ferenczy, G. G.; Kállay, M. Exact density functional and wave function embedding schemes based on orbital localization.*J. Chem. Phys.*2016,*145*, 064107 DOI: 10.1063/1.4960177Google Scholar104https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xhtlalu7%252FP&md5=4e1dbf60af141a2d3e9e0d10a5ced940Exact density functional and wave function embedding schemes based on orbital localizationHegely, Bence; Nagy, Peter R.; Ferenczy, Gyorgy G.; Kallay, MihalyJournal of Chemical Physics (2016), 145 (6), 064107/1-064107/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Exact schemes for the embedding of d. functional theory (DFT) and wave function theory (WFT) methods into lower-level DFT or WFT approaches are introduced utilizing orbital localization. First, a simple modification of the projector-based embedding scheme of Manby and co-workers [J. Chem. Phys. 140, 18A507 (2014)] is proposed. We also use localized orbitals to partition the system, but instead of augmenting the Fock operator with a somewhat arbitrary level-shift projector we solve the Huzinaga-equation, which strictly enforces the Pauli exclusion principle. Second, the embedding of WFT methods in local correlation approaches is studied. Since the latter methods split up the system into local domains, very simple embedding theories can be defined if the domains of the active subsystem and the environment are treated at a different level. The considered embedding schemes are benchmarked for reaction energies and compared to quantum mechanics (QM)/mol. mechanics (MM) and vacuum embedding. We conclude that for DFT-in-DFT embedding, the Huzinaga-equation-based scheme is more efficient than the other approaches, but QM/MM or even simple vacuum embedding is still competitive in particular cases. Concerning the embedding of wave function methods, the clear winner is the embedding of WFT into low-level local correlation approaches, and WFT-in-DFT embedding can only be more advantageous if a non-hybrid d. functional is employed. (c) 2016 American Institute of Physics.**105**Polly, R.; Werner, H.-J.; Manby, F. R.; Knowles, P. J. Fast Hartree-Fock theory using local fitting approximations.*Mol. Phys.*2004,*102*, 2311, DOI: 10.1080/0026897042000274801Google Scholar105https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXhtFGmu73I&md5=65b7443d485e77025637ed658a45d16eFast Hartree-Fock theory using local density fitting approximationsPolly, Robert; Werner, Hans-Joachim; Manby, Frederick R.; Knowles, Peter J.Molecular Physics (2004), 102 (21-22), 2311-2321CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)D. fitting approxns. are applied to generate the Fock matrix in Hartree-Fock calcns. By localizing the orbitals in each iteration and performing sep. fits for each orbital the O(N4) scaling of the computational effort for the exchange can be reduced to O(N). We also use the Poisson method to replace almost all Coulomb integrals with simple overlaps, an efficient alternative to diagonalization, and dual basis sets such that the Hartree-Fock calcn. is performed in a smaller basis than the subsequent treatment of electron correlation. The accuracy and efficiency of the method is demonstrated in calcns. with almost 4000 basis functions. The errors introduced by the local approxns. on HF and MP2 energies are small compared to those that arise from the d. fitting, and the fitting errors themselves (typically 1-10 microhartree per atom) are very small compared, for example, to the effect of basis set variations.**106**Köppl, C.; Werner, H.-J. Parallel and Low-Order Scaling Implementation of Hartree-Fock Exchange Using Local Density Fitting.*J. Chem. Theory Comput.*2016,*12*, 3122, DOI: 10.1021/acs.jctc.6b00251Google Scholar106https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC2s%252Fnt1Wksg%253D%253D&md5=b3bdd622dd764ccb35bfcef7e0d2efbbParallel and Low-Order Scaling Implementation of Hartree-Fock Exchange Using Local Density FittingKoppl Christoph; Werner Hans-JoachimJournal of chemical theory and computation (2016), 12 (7), 3122-34 ISSN:.Calculations using modern linear-scaling electron-correlation methods are often much faster than the necessary reference Hartree-Fock (HF) calculations. We report a newly implemented HF program that speeds up the most time-consuming step, namely, the evaluation of the exchange contributions to the Fock matrix. Using localized orbitals and their sparsity, local density fitting (LDF), and atomic orbital domains, we demonstrate that the calculation of the exchange matrix scales asymptotically linearly with molecular size. The remaining parts of the HF calculation scale cubically but become dominant only for very large molecular sizes or with many processing cores. The method is well parallelized, and the speedup scales well with up to about 100 CPU cores on multiple compute nodes. The effect of the local approximations on the accuracy of computed HF and local second-order Moller-Plesset perturbation theory energies is systematically investigated, and default values are established for the parameters that determine the domain sizes. Using these values, calculations for molecules with hundreds of atoms in combination with triple-ζ basis sets can be carried out in less than 1 h, with just a few compute nodes. The method can also be used to speed up density functional theory calculations with hybrid functionals that contain HF exchange.**107**Csóka, J.; Kállay, M. Speeding up density fitting Hartree-Fock calculations with multipole approximations.*Mol. Phys.*2020,*118*, e1769213 DOI: 10.1080/00268976.2020.1769213Google Scholar107https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhsVGlsb%252FF&md5=d4d896ec3424563948203bb2983ca4a6Speeding up density fitting Hartree-Fock calculations with multipole approximationsCsoka, Jozsef; Kallay, MihalyMolecular Physics (2020), 118 (19-20), e1769213/1-e1769213/16CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)The multipole approxn. is utilized for reducing the computational expenses of the exchange contribution in d. fitting Hartree-Fock (DF-HF) calcns. Strategies for approximating the relevant three-center Coulomb integrals with the multipole expansion are discussed. Based on the factorised form of the integrals, an algorithm is proposed for the evaluation of the exchange term for both conventional and local DF-HF methods. The accuracy of the resulting energies, the numerical stability of the algorithm, and the speedups achieved are benchmarked with respect to the order of the multipole expansion for various mol. systems. Our results suggest that computation times for a conventional DF-HF calcn. can be reduced roughly by a factor of 1.5 for mols. of a couple of hundreds of atoms without any loss of accuracy, while the speedups are somewhat more moderate if local d. fitting approxns. are also deployed.**108**Csóka, J.; Kállay, M. Speeding up Hartree-Fock and Kohn-Sham calculations with first-order corrections*J. Chem. Phys.*2021, 154. submitted.Google ScholarThere is no corresponding record for this reference.**109**Foster, J. M.; Boys, S. F. Canonical Configurational Interaction Procedure.*Rev. Mod. Phys.*1960,*32*, 300, DOI: 10.1103/RevModPhys.32.300Google Scholar109https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaF3cXht1Cnsbw%253D&md5=abc42703f7689c501515853f76db14a6Canonical configurational interaction procedureFoster, J.M.; Boys, S. F.Reviews of Modern Physics (1960), 32 (), 300-2CODEN: RMPHAT; ISSN:0034-6861.A method of choosing predetor functions to express wave functions in their briefest form is proposed. Besides facilitating calcns., this choice of functions appears to approx. chem. invariant orbitals. The method is restricted to electronic states in which a crude approxn. can be obtained in the form of a single Slater determinant.**110**Pipek, J.; Mezey, P. A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functions.*J. Chem. Phys.*1989,*90*, 4916, DOI: 10.1063/1.456588Google Scholar110https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1MXks1Cht7w%253D&md5=c983656b61c0ec520ce20cd8773f87c6A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functionsPipek, Janos; Mezey, Paul G.Journal of Chemical Physics (1989), 90 (9), 4916-26CODEN: JCPSA6; ISSN:0021-9606.A new intrinsic localization algorithm is suggested based on a recently developed math. measure of localization. No external criteria are used to define a priori bonds, lone pairs, and core orbitals. The method similarly to Edmiston-Ruedenberg's localization prefers the well established chem. concept of σ-π sepn., while on the other hand, works as economically as Boys' procedure. For the applications of the new localization algorithm, no addnl. quantities are to be calcd., the knowledge of at. overlap integrals is sufficient. This feature allows a unique formulation of the theory, adaptable for both ab initio and semiempirical methods, even in those cases where the exact form of the at. basis functions is not defined (line in the EHT and PPP calcns). The implementation of the procedure in already existing program systems is particularly easy. The Emiston-Ruedenberg and Boys localized orbitals are compared with those calcd. by the method suggested here, within both the CNDO/2 and ab initio frameworks (using STO-3G and 6-31G** basis sets) for several mols. (CO, H2CO, B2H6, and N2O4).**111**Boughton, J. W.; Pulay, P. Comparison of the Boys and Pipek-Mezey Localizations in the Local Correlation Approach and Automatic Virtual Basis Selection.*J. Comput. Chem.*1993,*14*, 736, DOI: 10.1002/jcc.540140615Google Scholar111https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXlt1Gntbc%253D&md5=7a43c82108697943ba41bb8cd6363d9cComparisons of the Boys and Pipek-Mezey localizations in the local correlation approach and automatic virtual basis selectionBoughton, James W.; Pulay, PeterJournal of Computational Chemistry (1993), 14 (6), 736-40CODEN: JCCHDD; ISSN:0192-8651.The authors' implementation of Pipek-Mezey electron population localization is described. It is compared with other localization schemes, and its use in the framework of the local-correlation method is discussed. For such use, this localization is shown to be clearly superior to the Boys localization method in the case of phys. well-localized systems. The authors' current algorithm for selection of local virtual spaces is also described.**112**Nagy, P. R.; Surján, P. R.; Szabados, Á. Mayer’s orthogonalization: relation to the Gram-Schmidt and Löwdin’s symmetrical scheme.*Theor. Chem. Acc.*2012,*131*, 1109, DOI: 10.1007/s00214-012-1109-yGoogle ScholarThere is no corresponding record for this reference.**113**Tóth, Z.; Nagy, P. R.; Jeszenszki, P.; Szabados, Á. Novel orthogonalization and biorthogonalization algorithms.*Theor. Chem. Acc.*2015,*134*, 100, DOI: 10.1007/s00214-015-1703-xGoogle ScholarThere is no corresponding record for this reference.**114**Boys, S. F.; Cook, G. B.; Reeves, C. M.; Shavitt, I. Automatic Fundamental Calculations of Molecular Structure.*Nature*1956,*178*, 1207, DOI: 10.1038/1781207a0Google ScholarThere is no corresponding record for this reference.**115**Whitten, J. L. Coulombic potential energy integrals and approximations.*J. Chem. Phys.*1973,*58*, 4496, DOI: 10.1063/1.1679012Google Scholar115https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE3sXktFSjtbo%253D&md5=05c510c8db660386b6fd3df789c22b10Coulombic potential energy integrals and approximationsWhitten, J. L.Journal of Chemical Physics (1973), 58 (10), 4496-501CODEN: JCPSA6; ISSN:0021-9606.Theorems are derived which establish a method of approxg. 2-particle Coulombic potential energy integrals, [.vphi.a(1)|r12-1|.vphi.b-(2)], in terms of approx. charge ds. .vphi.a' and .vphi.b'. Rigorous error bounds, |[.vphi.a(1)|r12-1|.vphi.b(2)] - [.vphi.a'(1)|r12-1|.vphi.b'(2)]| ≤ δ, are simply expressed in terms of information calcd. sep. for the pair of ds. .vphi.a and .vphi.b' and the pair .vphi.b and .vphi.b'. From the structure of the bound, a simple method of optimizing charge d. approxns. such that δ is minimized is derived. The framework of the theory appears to be well suited for application to the approxn. of electron repulsion integrals which occur in mol. structure theory, and applications to the approxn. of integrals over Slater orbitals or grouped Gaussian functions are discussed.**116**Samu, G.; Kállay, M. Efficient evaluation of three-center Coulomb intergrals.*J. Chem. Phys.*2017,*146*, 204101 DOI: 10.1063/1.4983393Google Scholar116https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXotlWit7o%253D&md5=bc9ec710ebda93e61763d91d75c57e92Efficient evaluation of three-center Coulomb integralsSamu, Gyula; Kallay, MihalyJournal of Chemical Physics (2017), 146 (20), 204101/1-204101/19CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)In this study we pursue the most efficient paths for the evaluation of three-center electron repulsion integrals (ERIs) over solid harmonic Gaussian functions of various angular momenta. First, the adaptation of the well-established techniques developed for four-center ERIs, such as the Obara-Saika, McMurchie-Davidson, Gill-Head-Gordon-Pople, and Rys quadrature schemes, and the combinations thereof for three-center ERIs is discussed. Several algorithmic aspects, such as the order of the various operations and primitive loops as well as pre-screening strategies, are analyzed. Second, the no. of floating point operations (FLOPs) is estd. for the various algorithms derived, and based on these results the most promising ones are selected. We report the efficient implementation of the latter algorithms invoking automated programming techniques and also evaluate their practical performance. We conclude that the simplified Obara-Saika scheme of Ahlrichs is the most cost-effective one in the majority of cases, but the modified Gill-Head-Gordon-Pople and Rys algorithms proposed herein are preferred for particular shell triplets. Our numerical expts. also show that even though the solid harmonic transformation and the horizontal recurrence require significantly fewer FLOPs if performed at the contracted level, this approach does not improve the efficiency in practical cases. Instead, it is more advantageous to carry out these operations at the primitive level, which allows for more efficient integral pre-screening and memory layout. (c) 2017 American Institute of Physics.**117**Kállay, M. A systematic way for the cost reduction of density fitting methods.*J. Chem. Phys.*2014,*141*, 244113 DOI: 10.1063/1.4905005Google Scholar117https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXitFymsbnK&md5=721f8e1df5009126080ebf5d389f545aA systematic way for the cost reduction of density fitting methodsKallay, MihalyJournal of Chemical Physics (2014), 141 (24), 244113/1-244113/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present a simple approach for the redn. of the size of auxiliary basis sets used in methods exploiting the d. fitting (resoln. of identity) approxn. for electron repulsion integrals. Starting out of the singular value decompn. of three-center two-electron integrals, new auxiliary functions are constructed as linear combinations of the original fitting functions. The new functions, which we term natural auxiliary functions (NAFs), are analogous to the natural orbitals widely used for the cost redn. of correlation methods. The use of the NAF basis enables the systematic truncation of the fitting basis, and thereby potentially the redn. of the computational expenses of the methods, though the scaling with the system size is not altered. The performance of the new approach has been tested for several quantum chem. methods. It is demonstrated that the most pronounced gain in computational efficiency can be expected for iterative models which scale quadratically with the size of the fitting basis set, such as the direct RPA. The approach also has the promise of accelerating local correlation methods, for which the processing of three-center Coulomb integrals is a bottleneck. (c) 2014 American Institute of Physics.**118**Gyevi-Nagy, L.; Kállay, M.; Nagy, P. R. Accurate reduced-cost CCSD(T) energies: parallel implementation, benchmarks, and large-scale applications.*J. Chem. Theory Comput.*2021,*17*, 860, DOI: 10.1021/acs.jctc.0c01077Google Scholar118https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXislWkug%253D%253D&md5=e324434e3811d234dc2075f351d7c3d5Accurate Reduced-Cost CCSD(T) Energies: Parallel Implementation, Benchmarks, and Large-Scale ApplicationsGyevi-Nagy, Laszlo; Kallay, Mihaly; Nagy, Peter R.Journal of Chemical Theory and Computation (2021), 17 (2), 860-878CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The accurate and systematically improvable frozen natural orbital (FNO) and natural auxiliary function (NAF) cost-reducing approaches are combined with our recent coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] implementations. Both of the closed- and open-shell FNO-CCSD(T) codes benefit from OpenMP parallelism, completely or partially integral-direct d.-fitting algorithms, checkpointing, and hand-optimized, memory- and operation count effective implementations exploiting all permutational symmetries. The closed-shell CCSD(T) code requires negligible disk I/O and network bandwidth, is MPI/OpenMP parallel, and exhibits outstanding peak performance utilization of 50-70% up to hundreds of cores. Conservative FNO and NAF truncation thresholds benchmarked for challenging reaction, atomization, and ionization energies of both closed- and open-shell species are shown to maintain 1 kJ/mol accuracy against canonical CCSD(T) for systems of 31-43 atoms even with large basis sets. The cost redn. of up to an order of magnitude achieved extends the reach of FNO-CCSD(T) to systems of 50-75 atoms (up to 2124 AOs) with triple- and quadruple-ζ basis sets, which is unprecedented without local approxns. Consequently, a considerably larger portion of the chem. compd. space can now be covered by the practically "gold std." quality FNO-CCSD(T) method using affordable resources and about a week of wall time. Large-scale applications are presented for organo-catalytic and transition-metal reactions as well as noncovalent interactions. Possible applications for benchmarking local CCSD(T) methods, as well as for the accuracy assessment or parametrization of less complete models, for example, d. functional approxns. or machine learning potentials, are also outlined.**119**Graham, D. C.; Menon, A. S.; Goerigk, L.; Grimme, S.; Radom, L. Optimization and Basis-Set Dependence of a Restricted-Open-Shell Form of B2-PLYP Double-Hybrid Density Functional Theory.*J. Phys. Chem. A*2009,*113*, 9861, DOI: 10.1021/jp9042864Google Scholar119https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXpt1Kjsrs%253D&md5=671d36c37190c3d50eb5e7119ed10155Optimization and Basis-Set Dependence of a Restricted-Open-Shell Form of B2-PLYP Double-Hybrid Density Functional TheoryGraham, David C.; Menon, Ambili S.; Goerigk, Lars; Grimme, Stefan; Radom, LeoJournal of Physical Chemistry A (2009), 113 (36), 9861-9873CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)The performance of the restricted-open-shell form of the double-hybrid d. functional theory (DHDFT) B2-PLYP procedure has been compared with that of its unrestricted counterpart using the G3/05 test set. Addnl., the influence of basis set on the parametrization and performance of ROB2-PLYP, and the further improvement of ROB2-PLYP through augmentation with a long-range dispersion function, have been investigated. We find that, after optimization of the two empirical DHDFT parameters, the ROB2-PLYP method (HF exchange = 59% and MP2 correlation = 28%) performs slightly better than the corresponding UB2-PLYP method (HF exchange = 62% and MP2 correlation = 35%), with mean abs. deviations (MADs) from the exptl. energies in the G3/05 test set of 9.1 and 9.9 kJ mol-1, resp., when the cc-pVQZ basis set is employed. Sep. optimizations of the parameters for the RO and U procedures are crucial for a fair comparison. For example, for the G2/97 test set, ROB2-PLYP(53,27) and ROB2-PLYP(62,35) show MADs of 12.2 and 13.5 kJ mol-1, resp., compared with the 6.6 kJ mol-1 for (the optimized) ROB2-PLYP(59,28). The performance of ROB2-PLYP deteriorates significantly as the basis-set size is decreased, reflecting the enhanced basis-set dependence of the MP2 contribution compared with std. DFT. We find that this deficiency can be partly overcome through reparametrization. However, when the basis set drops below triple-ζ, the improvements made on reoptimizing the ROB2-PLYP parameters are not sufficient to warrant their general use. We find that the dispersion- and BSSE-cor. ROB2-PLYP(59,28)-D HCP procedure performs significantly better than ROB2-PLYP(59,28) for the S22 test set of interaction energies in which dispersion interactions are particularly important, with the MAD falling from 6.1 to 1.6 kJ mol-1. However, when the same D correction is applied to the G3/05 test set, the performance of ROB2-PLYP(59,28)-D deteriorates slightly compared with ROB2-PLYP(59,28), with the MAD increasing from 9.1 to 9.5 kJ mol-1.**120**Guo, Y.; Sivalingam, K.; Valeev, E. F.; Neese, F. SparseMaps—A systematic infrastructure for reduced-scaling electronic structure methods. III. Linear-scaling multireference domain-based pair natural orbital N-electron valence perturbation theory.*J. Chem. Phys.*2016,*144*, 094111 DOI: 10.1063/1.4942769Google Scholar120https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XktVWkur0%253D&md5=8d8ba6540a8765d30b2005f7f141642fSparseMaps-A systematic infrastructure for reduced-scaling electronic structure methods. III. Linear-scaling multireference domain-based pair natural orbital N-electron valence perturbation theoryGuo, Yang; Sivalingam, Kantharuban; Valeev, Edward F.; Neese, FrankJournal of Chemical Physics (2016), 144 (9), 094111/1-094111/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Multi-ref. (MR) electronic structure methods, such as MR CI or MR perturbation theory, can provide reliable energies and properties for many mol. phenomena like bond breaking, excited states, transition states or magnetic properties of transition metal complexes and clusters. However, owing to their inherent complexity, most MR methods are still too computationally expensive for large systems. Therefore the development of more computationally attractive MR approaches is necessary to enable routine application for large-scale chem. systems. Among the state-of-the-art MR methods, second-order N-electron valence state perturbation theory (NEVPT2) is an efficient, size-consistent, and intruder-state-free method. However, there are still two important bottlenecks in practical applications of NEVPT2 to large systems: (a) the high computational cost of NEVPT2 for large mols., even with moderate active spaces and (b) the prohibitive cost for treating large active spaces. In this work, we address problem (a) by developing a linear scaling "partially contracted" NEVPT2 method. This development uses the idea of domain-based local pair natural orbitals (DLPNOs) to form a highly efficient algorithm. As shown previously in the framework of single-ref. methods, the DLPNO concept leads to an enormous redn. in computational effort while at the same time providing high accuracy (approaching 99.9% of the correlation energy), robustness, and black-box character. In the DLPNO approach, the virtual space is spanned by pair natural orbitals that are expanded in terms of projected AOs in large orbital domains, while the inactive space is spanned by localized orbitals. The active orbitals are left untouched. Our implementation features a highly efficient "electron pair prescreening" that skips the negligible inactive pairs. The surviving pairs are treated using the partially contracted NEVPT2 formalism. A detailed comparison between the partial and strong contraction schemes is made, with conclusions that discourage the strong contraction scheme as a basis for local correlation methods due to its non-invariance with respect to rotations in the inactive and external subspaces. A minimal set of conservatively chosen truncation thresholds controls the accuracy of the method. With the default thresholds, about 99.9% of the canonical partially contracted NEVPT2 correlation energy is recovered while the crossover of the computational cost with the already very efficient canonical method occurs reasonably early; in linear chain type compds. at a chain length of around 80 atoms. Calcns. are reported for systems with more than 300 atoms and 5400 basis functions. (c) 2016 American Institute of Physics.**121**Kállay, M.; Nagy, P. R.; Mester, D.; Rolik, Z.; Samu, G.; Csontos, J.; Csóka, J.; Szabó, P. B.; Gyevi-Nagy, L.; Hégely, B.; Ladjánszki, I.; Szegedy, L.; Ladóczki, B.; Petrov, K.; Farkas, M.; Mezei, P. D.; Ganyecz, Á. The MRCC program system: Accurate quantum chemistry from water to proteins.*J. Chem. Phys.*2020,*152*, 074107 DOI: 10.1063/1.5142048Google Scholar121https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXjs1Ogs70%253D&md5=d57c0d168fa449d482a0ded4ef46dd3dThe MRCC program system: Accurate quantum chemistry from water to proteinsKallay, Mihaly; Nagy, Peter R.; Mester, David; Rolik, Zoltan; Samu, Gyula; Csontos, Jozsef; Csoka, Jozsef; Szabo, P. Bernat; Gyevi-Nagy, Laszlo; Hegely, Bence; Ladjanszki, Istvan; Szegedy, Lorant; Ladoczki, Bence; Petrov, Klara; Farkas, Mate; Mezei, Pal D.; Ganyecz, AdamJournal of Chemical Physics (2020), 152 (7), 074107CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)MRCC is a package of ab initio and d. functional quantum chem. programs for accurate electronic structure calcns. The suite has efficient implementations of both low- and high-level correlation methods, such as second-order Moller-Plesset (MP2), RPA, second-order algebraic-diagrammatic construction [ADC(2)], coupled-cluster (CC), CI, and related techniques. It has a state-of-the-art CC singles and doubles with perturbative triples [CCSD(T)] code, and its specialties, the arbitrary-order iterative and perturbative CC methods developed by automated programming tools, enable achieving convergence with regard to the level of correlation. The package also offers a collection of multi-ref. CC and CI approaches. Efficient implementations of d. functional theory (DFT) and more advanced combined DFT-wave function approaches are also available. Its other special features, the highly competitive linear-scaling local correlation schemes, allow for MP2, RPA, ADC(2), CCSD(T), and higher-order CC calcns. for extended systems. Local correlation calcns. can be considerably accelerated by multi-level approxns. and DFT-embedding techniques, and an interface to mol. dynamics software is provided for quantum mechanics/mol. mechanics calcns. All components of MRCC support shared-memory parallelism, and multi-node parallelization is also available for various methods. For academic purposes, the package is available free of charge. (c) 2020 American Institute of Physics.**122**Kállay, M.; Nagy, P. R.; Rolik, Z.; Mester, D.; Samu, G.; Csontos, J.; Csóka, J.; Szabó, P. B.; Gyevi-Nagy, L.; Ladjánszki, I.; Szegedy, L.; Ladóczki, B.; Petrov, K.; Farkas, M.; Mezei, P. D.; Hégely, B. MRCC: A Quantum Chemical Program Suite. https://www.mrcc.hu/ (accessed Jan 1, 2021).Google ScholarThere is no corresponding record for this reference.**123**Weigend, F.; Ahlrichs, R. Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy integrals over Gaussian functions.*Phys. Chem. Chem. Phys.*2005,*7*, 3297, DOI: 10.1039/b508541aGoogle Scholar123https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXpsFWgu7o%253D&md5=a820fb6055c993b50c405ba0fc62b194Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracyWeigend, Florian; Ahlrichs, ReinhartPhysical Chemistry Chemical Physics (2005), 7 (18), 3297-3305CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)Gaussian basis sets of quadruple zeta valence quality for Rb-Rn are presented, as well as bases of split valence and triple zeta valence quality for H-Rn. The latter were obtained by (partly) modifying bases developed previously. A large set of more than 300 mols. representing (nearly) all elements-except lanthanides-in their common oxidn. states was used to assess the quality of the bases all across the periodic table. Quantities investigated were atomization energies, dipole moments and structure parameters for Hartree-Fock, d. functional theory and correlated methods, for which we had chosen Moller-Plesset perturbation theory as an example. Finally recommendations are given which type of basis set is used best for a certain level of theory and a desired quality of results.**124**Dunning, T. H., Jr. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen.*J. Chem. Phys.*1989,*90*, 1007, DOI: 10.1063/1.456153Google Scholar124https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1MXksVGmtrk%253D&md5=c6cd67a3748dc61692a9cb622d2694a0Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogenDunning, Thom H., Jr.Journal of Chemical Physics (1989), 90 (2), 1007-23CODEN: JCPSA6; ISSN:0021-9606.Guided by the calcns. on oxygen in the literature, basis sets for use in correlated at. and mol. calcns. were developed for all of the first row atoms from boron through neon, and for hydrogen. As in the oxygen atom calcns., the incremental energy lowerings, due to the addn. of correlating functions, fall into distinct groups. This leads to the concept of correlation-consistent basis sets, i.e., sets which include all functions in a given group as well as all functions in any higher groups. Correlation-consistent sets are given for all of the atoms considered. The most accurate sets detd. in this way, [5s4p3d2f1g], consistently yield 99% of the correlation energy obtained with the corresponding at.-natural-orbital sets, even though the latter contains 50% more primitive functions and twice as many primitive polarization functions. It is estd. that this set yields 94-97% of the total (HF + 1 + 2) correlation energy for the atoms neon through boron.**125**Dunning, T. H., Jr.; Peterson, K. A.; Wilson, A. K. Gaussian basis sets for use in correlated molecular calculations. X. The atoms aluminum through argon revisited.*J. Chem. Phys.*2001,*114*, 9244, DOI: 10.1063/1.1367373Google Scholar125https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3MXjvFWiurk%253D&md5=85d9d30757400eff4874e04cb67bcf2aGaussian basis sets for use in correlated molecular calculations. X. The atoms aluminum through argon revisitedDunning, Thom H., Jr.; Peterson, Kirk A.; Wilson, Angela K.Journal of Chemical Physics (2001), 114 (21), 9244-9253CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)For mols. contg. second row atoms, unacceptable errors have been found in extrapolating dissocn. energies calcd. with the std. correlation consistent basis sets to the complete basis set limit. By carefully comparing the convergence behavior of De(O2) and De(SO), we show that the cause of these errors is a result of two inter-related problems: near duplication of the exponents in two of the d sets and a lack of high-exponent functions in the early members of the sets. Similar problems exist for the f sets (and probably in higher angular momentum sets), but have only a minor effect on the calcd. dissocn. energies. A no. of approaches to address the problems in the d sets were investigated. Well behaved convergence was obtained by augmenting the (1d) and (2d) sets with a high-exponent function and by replacing the (3d) set by the (4d) set and the (4d) set by the (5d) set and so on. To ensure satisfactory coverage of both the L and M shell regions, the exponents of the new d sets were re-optimized. Benchmark calcns. on Si2, PN, SO, and AlCl with the new cc-pV(n + d)Z sets show greatly improved convergence behavior not only for De but for many other properties as well.**126**Weigend, F.; Köhn, A.; Hättig, C. Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations.*J. Chem. Phys.*2002,*116*, 3175, DOI: 10.1063/1.1445115Google Scholar126https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XhtlSiu7k%253D&md5=0130fa656254a693e80d4be6b0f442b8Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculationsWeigend, Florian; Kohn, Andreas; Hattig, ChristofJournal of Chemical Physics (2002), 116 (8), 3175-3183CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The convergence of the second-order Moller-Plesset perturbation theory (MP2) correlation energy with the cardinal no. X is investigated for the correlation consistent basis-set series cc-pVXZ and cc-pV(X+d)Z. For the aug-cc-pVXZ and aug-cc-pV(X+d)Z series the convergence of the MP2 correlation contribution to the dipole moment is studied. It is found that, when d-shell electrons cannot be frozen, the cc-pVXZ and aug-cc-pVXZ basis sets converge much slower for third-row elements then they do for first- and second-row elements. Based on the results of these studies criteria are deduced for the accuracy of auxiliary basis sets used in the resoln. of the identity (RI) approxn. for electron repulsion integrals. Optimized auxiliary basis sets for RI-MP2 calcns. fulfilling these criteria are reported for the sets cc-pVXZ, cc-pV(X+d)Z, aug-cc-pVXZ, and aug-cc-pV(X+d)Z with X=D, T, and Q. For all basis sets the RI error in the MP2 correlation energy is more than two orders of magnitude smaller than the usual basis-set error. For the auxiliary aug-cc-pVXZ and aug-cc-pV(X+d)Z sets the RI error in the MP2 correlation contribution to the dipole moment is one order of magnitude smaller than the usual basis set error. Therefore extrapolations towards the basis-set limit are possible within the RI approxn. for both energies and properties. The redn. in CPU time obtained with the RI approxn. increases rapidly with basis set size. For the cc-pVQZ basis an acceleration by a factor of up to 170 is obsd.**127**Karton, A.; Martin, J. M. L. Comment on: “Estimating the Hartree-Fock limit from finite basis set calculations”.*Theor. Chem. Acc.*2006,*115*, 330, DOI: 10.1007/s00214-005-0028-6Google Scholar127https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28Xjt1yksbY%253D&md5=c460b4d0cd4aa81e4471bbdd2ba4b3c1Comment on: "Estimating the Hartree-Fock limit from finite basis set calculations" [Jensen F (2005) Theor Chem Acc 113:267]Karton, Amir; Martin, Jan M. L.Theoretical Chemistry Accounts (2006), 115 (4), 330-333CODEN: TCACFW; ISSN:1432-881X. (Springer GmbH)We demonstrate that a minor modification of the extrapolation proposed by Jensen [(2005): Theor Chem Acc 113: 267] yields very reliable ests. of the Hartree-Fock limit in conjunction with correlation consistent basis sets. Specifically, a two-point extrapolation of the form yields HF limits E HF,∞ with an RMS error of 0.1 millihartree using aug-cc-pVQZ and aug-cc-pV5Z basis sets, and of 0.01 millihartree using aug-cc-pV5Z and aug-cc-pV6Z basis sets.**128**Helgaker, T.; Klopper, W.; Koch, H.; Noga, J. Basis-set convergence of correlated calculations on water.*J. Chem. Phys.*1997,*106*, 9639, DOI: 10.1063/1.473863Google Scholar128https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXjvVCgu78%253D&md5=f4689c1b38fe30eb721e9cd7d607bdf7Basis-set convergence of correlated calculations on waterHelgaker, Trygve; Klopper, Wim; Koch, Henrik; Noga, JozefJournal of Chemical Physics (1997), 106 (23), 9639-9646CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The basis-set convergence of the electronic correlation energy in the water mol. is investigated at the second-order Moller-Plesset level and at the coupled-cluster singles-and-doubles level with and without perturbative triples corrections applied. The basis-set limits of the correlation energy are established to within 2mEh by means of (1) extrapolations from sequences of calcns. using correlation-consistent basis sets and (2) from explicitly correlated calcns. employing terms linear in the inter-electronic distances rij. For the extrapolations to the basis-set limit of the correlation energies, fits of the form a + bX-3 (where X is two for double-zeta sets, three for triple-zeta sets, etc.) are found to be useful. CCSD(T) calcns. involving as many as 492 AOs are reported.**129**Goerigk, L.; Grimme, S. A general database for main group thermochemistry, kinetics, and noncovalent interactions—Assessment of common and reparameterized (meta-)GGA density functionals.*J. Chem. Theory Comput.*2010,*6*, 107, DOI: 10.1021/ct900489gGoogle Scholar129https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhsVClurvM&md5=6452b32bf508de27fb37c221b8fdfdd4A General Database for Main Group Thermochemistry, Kinetics, and Noncovalent Interactions - Assessment of Common and Reparameterized (meta-)GGA Density FunctionalsGoerigk, Lars; Grimme, StefanJournal of Chemical Theory and Computation (2010), 6 (1), 107-126CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a quantum chem. benchmark database for general main group thermochem., kinetics, and noncovalent interactions (GMTKN24). It is an unprecedented compilation of 24 different, chem. relevant subsets that either are taken from already existing databases or are presented here for the first time. The complete set involves a total of 1.049 at. and mol. single point calcns. and comprises 731 data points (relative chem. energies) based on accurate theor. or exptl. ref. values. The usefulness of the GMTKN24 database is shown by applying common d. functionals on the (meta-)generalized gradient approxn. (GGA), hybrid-GGA, and double-hybrid-GGA levels to it, including an empirical London dispersion correction. Furthermore, we refitted the functional parameters of four (meta-)GGA functionals based on a fit set contg. 143 systems, comprising seven chem. different problems. Validation against the GMTKN24 and the mol. structure (bond lengths) databases shows that the reparameterization does not change bond lengths much, whereas the description of energetic properties is more prone to the parameters' values. The empirical dispersion correction also often improves for conventional thermodn. problems and makes a functional's performance more uniform over the entire database. The refitted functionals typically have a lower mean abs. deviation for the majority of subsets in the proposed GMTKN24 set. This, however, is also often accompanied at the expense of poor performance for a few other important subsets. Thus, creating a broadly applicable (and overall better) functional by just reparameterizing existing ones seems to be difficult. Nevertheless, this benchmark study reveals that a reoptimized (i.e., empirical) version of the TPSS-D functional (oTPSS-D) performs well for a variety of problems and may meet the stds. of an improved functional. We propose validation against this new compilation of benchmark sets as a definitive way to evaluate a new quantum chem. method's true performance.**130**Liu, Y. Linear Scaling High-spin Open-shell Local Correlation Methods. Ph.D. Thesis, Institut für Theoretische Chemie der Universität Stuttgart, 2011.Google ScholarThere is no corresponding record for this reference.**131**Ghafarian Shirazi, R.; Neese, F.; Pantazis, D. A. Accurate Spin-State Energetics for Aryl Carbenes.*J. Chem. Theory Comput.*2018,*14*, 4733, DOI: 10.1021/acs.jctc.8b00587Google Scholar131https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXhsFWhurbE&md5=5a1ca0149cb41e07315bce58ca8be9f4Accurate Spin-State Energetics for Aryl CarbenesGhafarian Shirazi, Reza; Neese, Frank; Pantazis, Dimitrios A.Journal of Chemical Theory and Computation (2018), 14 (9), 4733-4746CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A test set of 12 aryl carbenes (AC12) is compiled with the purpose of establishing their adiabatic singlet-triplet energy splittings using correlated wave function based methods. The set covers both singlet and triplet ground state aryl carbenes, as well as a range of magnitudes for the ground state to excited state gap. The performance of coupled cluster methods is examd. with respect to the ref. wave function, the basis set, and a no. of addnl. methodol. parameters that enter the calcn. Inclusion of perturbative triples and basis set extrapolation with a combination of triple and quadruple-ζ basis sets are both required to ensure high accuracy. When canonical coupled cluster calcns. become too expensive, the domain-based local pair natural orbital approach DLPNO-CCSD(T) can be used as a reliable method for larger systems, as it achieves a mean abs. error of only 0.2 kcal/mol for the singlet-triplet gaps in the present test set. Other first-principles wave function methods and selected d. functional methods are also evaluated. Second-order Moller-Plesset perturbation theory approaches are only applicable in conjunction with orbital optimization (OO-MP2). Among the representative d. functional methods tested, only double hybrid functionals perform sufficiently accurately to be considered useful for systems with small singlet-triplet gaps. On the basis of the ref. coupled cluster results, projected gas-phase free energies are reported for all aryl carbenes. Finally, the treatment of singlet-triplet gaps in soln. is discussed in terms of both implicit and explicit solvation.**132**Wick, C. R.; Smith, D. M. Modeling the Reactions Catalyzed by Coenzyme B12 Dependent Enzymes: Accuracy and Cost-Quality Balance.*J. Phys. Chem. A*2018,*122*, 1747, DOI: 10.1021/acs.jpca.7b11798Google Scholar132https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXitVKrurw%253D&md5=43632b3bd8457f8670449e02bcea880bModeling the Reactions Catalyzed by Coenzyme B12 Dependent Enzymes: Accuracy and Cost-Quality BalanceWick, Christian R.; Smith, David M.Journal of Physical Chemistry A (2018), 122 (6), 1747-1755CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)The reactions catalyzed by coenzyme B12 dependent enzymes are formally initiated by the homolytic cleavage of a carbon-cobalt bond and a subsequent or concerted H-atom-transfer reaction. A reasonable model chem. for describing those reactions should, therefore, account for an accurate description of both reactions. The inherent limitation due to the necessary system size renders the coenzyme B12 system a suitable candidate for DFT or hybrid QM/MM methods; however, the accurate description of both homolytic Co-C cleavage and H-atom-transfer reactions within this framework is challenging and can lead to controversial results with varying accuracy. We present an assessment study of 16 common d. functionals applied to prototypical model systems for both reactions. H-abstraction reactions were modeled on the basis of four ref. reactions designed to resemble a broad range of coenzyme B12 reactions. The Co-C cleavage reaction is treated by an ONIOM(QM/MM) setup that is in excellent agreement with soln.-phase exptl. data and is as accurate as full DFT calcns. on the complete model system. We find that the meta-GGAs TPSS-D3 and M06L-D3 and the meta-hybrid M06-D3 give the best overall performance with MUEs for both types of reactions below 10 kJ mol-1. Our recommended model chem. allows for a fast and accurate description of coenzyme B12 chem. that is readily applicable to study the reactions in an enzymic framework.**133**Kiss, D. J.; Ferenczy, G. G. A detailed mechanism of the oxidative half-reaction of D-amino acid oxidase: another route for flavin oxidation.*Org. Biomol. Chem.*2019,*17*, 7973, DOI: 10.1039/C9OB00975BGoogle Scholar133https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhsFemtbjN&md5=8bdd81abd7503622b673b387abd4ae5fA detailed mechanism of the oxidative half-reaction of D-amino acid oxidase: another route for flavin oxidationKiss, Dora Judit; Ferenczy, Gyorgy G.Organic & Biomolecular Chemistry (2019), 17 (34), 7973-7984CODEN: OBCRAK; ISSN:1477-0520. (Royal Society of Chemistry)D-Amino acid oxidase (DAAO) is a flavoenzyme whose inhibition is expected to have therapeutic potential in schizophrenia. DAAO catalyzes hydride transfer from the substrate to the flavin in the reductive half-reaction, and the flavin is reoxidized by O2 in the oxidative half-reaction. Quantum mech./mol. mech. calcns. were performed and their results together with available exptl. information were used to elucidate the detailed mechanism of the oxidative half-reaction. The reaction starts with a single electron transfer from FAD to O2, followed by triplet-singlet transition. FAD oxidn. is completed by a proton coupled electron transfer to the oxygen species and the reaction terminates with H2O2 formation by proton transfer from the oxidized substrate to the oxygen species via a chain of water mols. The substrate plays a double role by facilitating the first electron transfer and by providing a proton in the last step. The mechanism differs from the oxidative half-reaction of other oxidases.**134**Paulechka, E.; Kazakov, A. Efficient Estimation of Formation Enthalpies for Closed-Shell Organic Compounds with Local Coupled-Cluster Methods.*J. Chem. Theory Comput.*2018,*14*, 5920, DOI: 10.1021/acs.jctc.8b00593Google Scholar134https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXhslGqs7jP&md5=3e6ee01852ef39366a9eb1abd23a92beEfficient Estimation of Formation Enthalpies for Closed-Shell Organic Compounds with Local Coupled-Cluster MethodsPaulechka, Eugene; Kazakov, AndreiJournal of Chemical Theory and Computation (2018), 14 (11), 5920-5932CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Efficient estn. of the enthalpies of formation for closed-shell org. compds. via atom-equiv.-type computational schemes and with the use of different local coupled-cluster with single, double, and perturbative triple excitation (CCSD(T)) approxns. was investigated. Detailed anal. of established sources of uncertainty, inclusive of contributions beyond frozen-core CCSD(T) and errors due to local CCSD(T) approxns. and zero-point energy anharmonicity, suggests the lower limit of about 2 kJ·mol-1 for the expanded uncertainty of the proposed estn. framework. Among the tested computational schemes, the best-performing cases demonstrate expanded uncertainty of about 2.5 kJ·mol-1, based on the anal. against 44 critically evaluated exptl. values. Computational efficiency, accuracy commensurable with that of a typical expt., and absence of the need for auxiliary reactions and addnl. exptl. data offer unprecedented advantages for practical use, such as prompt validation of existing measurements and estn. of missing values, as well as resoln. of exptl. conflicts. The utility of the proposed methodol. was demonstrated using a representative sample of the most recent exptl. measurements.**135**Sylvetsky, N.; Banerjee, A.; Alonso, M.; Martin, J. M. L. Performance of Localized Coupled Cluster Methods in a Moderately Strong Correlation Regime: Hückel-Möbius Interconversions in Expanded Porphyrins.*J. Chem. Theory Comput.*2020,*16*, 3641, DOI: 10.1021/acs.jctc.0c00297Google Scholar135https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXotFensbY%253D&md5=59abf3ac027c11aa7c977a8bef50ed82Performance of Localized Coupled Cluster Methods in a Moderately Strong Correlation Regime: Hueckel-Mobius Interconversions in Expanded PorphyrinsSylvetsky, Nitai; Banerjee, Ambar; Alonso, Mercedes; Martin, Jan M. L.Journal of Chemical Theory and Computation (2020), 16 (6), 3641-3653CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Localized orbital coupled cluster theory has recently emerged as a nonempirical alternative to DFT for large systems. Intuitively, one might expect such methods to perform less well for highly delocalized systems. In the present work, we apply both canonical CCSD(T) approxns. and a variety of localized approxns. to a set of flexible expanded porphyrins-macrocycles that can switch between Huckel, figure-eight, and Mobius topologies under external stimuli. Both min. and isomerization transition states are considered. We find that Mobius(-like) structures have much stronger static correlation character than the remaining structures, and that this causes significant errors in DLPNO-CCSD(T) and even DLPNO-CCSD(T1) approaches, unless TightPNO cutoffs are employed. If sub-kcal mol-1 accuracy with respect to canonical relative energies is required even for Mobius-type systems (or other systems plagued by strong static correlation), then Nagy and Kallay's LNO-CCSD(T) method with "tight" settings is the suitable localized approach. We propose the present POLYPYR21 data set as a benchmark for localized orbital methods or, more broadly, for the ability of lower-level methods to handle energetics with strongly varying degrees of static correlation.**136**Menezes, F.; Kats, D.; Werner, H.-J. Local complete active space second-order perturbation theory using pair natural orbitals (PNO-CASPT2).*J. Chem. Phys.*2016,*145*, 124115 DOI: 10.1063/1.4963019Google Scholar136https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhsF2ns73F&md5=18bcf033b22a13ac96e062ab5001d204Local complete active space second-order perturbation theory using pair natural orbitals (PNO-CASPT2)Menezes, Filipe; Kats, Daniel; Werner, Hans-JoachimJournal of Chemical Physics (2016), 145 (12), 124115/1-124115/20CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present a CASPT2 method which exploits local approxns. to achieve linear scaling of the computational effort with the mol. size, provided the active space is small and local. The inactive orbitals are localized, and the virtual space for each electron pair is spanned by a domain of pair-natural orbitals (PNOs). The configuration space is internally contracted, and the PNOs are defined for uniquely defined orthogonal pairs. Distant pair energies are obtained by multipole approxns., so that the no. of configurations that are explicitly treated in the CASPT2 scales linearly with mol. size (assuming a const. active space). The PNOs are generated using approx. amplitudes obtained in a pair-specific semi-canonical basis of projected AOs (PAOs). The evaluation and transformation of the two-electron integrals use the same parallel local d. fitting techniques as recently described for linear-scaling PNO-LMP2 (local second-order Moller-Plesset perturbation theory). The implementation of the amplitude equations, which are solved iteratively, employs the local integrated tensor framework. The efficiency and accuracy of the method are tested for excitation energies and correlation energies. It is demonstrated that the errors introduced by the local approxns. are very small. They can be well controlled by few parameters for the distant pair approxn., initial PAO domains, and the PNO domains. (c) 2016 American Institute of Physics.**137**Liakos, D. G.; Neese, F. Is It Possible To Obtain Coupled Cluster Quality Energies at near Density Functional Theory Cost? Domain-Based Local Pair Natural Orbital Coupled Cluster vs Modern Density Functional Theory.*J. Chem. Theory Comput.*2015,*11*, 4054, DOI: 10.1021/acs.jctc.5b00359Google Scholar137https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXht1agsr3F&md5=692fe9e72c609e18a489a9d919cdbefeIs It Possible To Obtain Coupled Cluster Quality Energies at near Density Functional Theory Cost? Domain-Based Local Pair Natural Orbital Coupled Cluster vs Modern Density Functional TheoryLiakos, Dimitrios G.; Neese, FrankJournal of Chemical Theory and Computation (2015), 11 (9), 4054-4063CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The recently developed domain-based local pair natural orbital coupled cluster theory with single, double, and perturbative triple excitations (DLPNO-CCSD(T)) delivers results that are closely approaching those of the parent canonical coupled cluster method at a small fraction of the computational cost. A recent extended benchmark study established that, depending on the three main truncation thresholds, it is possible to approach the canonical CCSD(T) results within 1 kJ (default setting, TightPNO), 1 kcal/mol (default setting, NormalPNO), and 2-3 kcal (default setting, LoosePNO). Although thresholds for calcns. with TightPNO are 2-4 times slower than those based on NormalPNO thresholds, they are still many orders of magnitude faster than canonical CCSD(T) calcns., even for small and medium sized mols. where there is little locality. The computational effort for the coupled cluster step scales nearly linearly with system size. Since, in many instances, the coupled cluster step in DLPNO-CCSD(T) is cheaper or at least not much more expensive than the preceding Hartree-Fock calcn., it is useful to compare the method against modern d. functional theory (DFT), which requires an effort comparable to that of Hartree-Fock theory (at least if Hartree-Fock exchange is part of the functional definition). Double hybrid d. functionals (DHDF's) even require a MP2-like step. The purpose of this article is to evaluate the cost vs accuracy ratio of DLPNO-CCSD(T) against modern DFT (including the PBE, B3LYP, M06-2X, B2PLYP, and B2GP-PLYP functionals and, where applicable, their van der Waals cor. counterparts). To eliminate any possible bias in favor of DLPNO-CCSD(T), we have chosen established benchmark sets that were specifically proposed for evaluating DFT functionals. It is demonstrated that DLPNO-CCSD(T) with any of the three default thresholds is more accurate than any of the DFT functionals. Furthermore, using the aug-cc-pVTZ basis set and the LoosePNO default settings, DLPNO-CCSD(T) is only about 1.2 times slower than B3LYP. With NormalPNO thresholds, DLPNO-CCSD(T) is about a factor of 2 slower than B3LYP and shows a mean abs. deviation of less than 1 kcal/mol to the ref. values for the four different data sets used. Our conclusion is that coupled cluster energies can indeed be obtained at near DFT cost.

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**1**Zhang, J.; Head-Gordon, M. Electronic structures and reaction dynamics of open-shell species.*Phys. Chem. Chem. Phys.*2009,*11*, 4699, DOI: 10.1039/b909815c1https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXms1yguro%253D&md5=d0ad9db7348abc93052b58e3f206afffElectronic structures and reaction dynamics of open-shell speciesZhang, Jingsong; Head-Gordon, MartinPhysical Chemistry Chemical Physics (2009), 11 (23), 4699-4700CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)There is no expanded citation for this reference.**2**Bally, T.; Borden, W. T.*Reviews in Computational Chemistry*; John Wiley & Sons, Ltd, 1999; pp 1– 97.There is no corresponding record for this reference.**3**Helgaker, T.; Jørgensen, P.; Olsen, J.*Molecular Electronic Structure Theory*; Wiley: Chichester, 2000.There is no corresponding record for this reference.**4**Krylov, A. I.*Reviews in Computational Chemistry*; John Wiley & Sons, Ltd, 2017; Chapter 4, pp 151– 224.There is no corresponding record for this reference.**5**Stanton, J. F.; Gauss, J.*Advances in Chemical Physics*; John Wiley & Sons, Ltd, 2003; pp 101– 146.There is no corresponding record for this reference.**6**Møller, C.; Plesset, M. S. Note on an Approximation Treatment for Many-Electron Systems.*Phys. Rev.*1934,*46*, 618, DOI: 10.1103/PhysRev.46.6186https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaA2MXnvVCq&md5=83bc74c244c7b18732ec9a47860d6e01Note on the approximation treatment for many-electron systemsMoller, Chr.; Plesset, M. S.Physical Review (1934), 46 (), 618-22CODEN: PHRVAO; ISSN:0031-899X.Math.**7**Raghavachari, K.; Trucks, G. W.; Pople, J. A.; Head-Gordon, M. A fifth-order perturbation comparison of electron correlation theories.*Chem. Phys. Lett.*1989,*157*, 479, DOI: 10.1016/S0009-2614(89)87395-67https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1MXlsVSkt7s%253D&md5=da2b01a30a044c9a1abcdfef4736aa1fA fifth-order perturbation comparison of electron correlation theoriesRaghavachari, Krishnan; Trucks, Gary W.; Pople, John A.; Head-Gordon, MartinChemical Physics Letters (1989), 157 (6), 479-83CODEN: CHPLBC; ISSN:0009-2614.Electron correlation theories such as CI (CI), coupled-cluster theory (CC), and quadratic CI (QCI) are assessed by means of a Moller-Plesset perturbation expansion of the correlation energy up to fifth order. The computational efficiencies and relative merits of the different techniques are outlined. A new augmented version of coupled-cluster theory, denoted as CCSD(T), is proposed to remedy some of the deficiencies of previous augmented coupled-cluster models.**8**Bartlett, R. J.; Musiał, M. Coupled-cluster theory in quantum chemistry.*Rev. Mod. Phys.*2007,*79*, 291, DOI: 10.1103/RevModPhys.79.2918https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXmt1Cqtbw%253D&md5=59fd2f595def41752de72a92c8ac510cCoupled-cluster theory in quantum chemistryBartlett, Rodney J.; Musial, MonikaReviews of Modern Physics (2007), 79 (1), 291-352CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)A review. Today, coupled-cluster theory offers the most accurate results among the practical ab initio electronic-structure theories applicable to moderate-sized mols. Though it was originally proposed for problems in physics, it has seen its greatest development in chem., enabling an extensive range of applications to mol. structure, excited states, properties, and all kinds of spectroscopy. In this review, the essential aspects of the theory are explained and illustrated with informative numerical results.**9**Cremer, D. M. Møller-Plesset perturbation theory: from small molecule methods to methods for thousands of atoms.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2011,*1*, 509, DOI: 10.1002/wcms.589https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXns1Wlt7Y%253D&md5=d8c9f6fea0f6ae7cbbc6631fabe8b2e8Moller-Plesset perturbation theory: From small molecule methods to methods for thousands of atomsCremer, DieterWiley Interdisciplinary Reviews: Computational Molecular Science (2011), 1 (4), 509-530CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)A review. The development of Mueller-Plesset perturbation theory (MPPT) has seen four different periods in almost 80 years. In the first 40 years (period 1), MPPT was largely ignored because the focus of quantum chemists was on variational methods. After the development of many-body perturbation theory by theor. physicists in the 1950s and 1960s, a second 20-yr long period started, during which MPn methods up to order n = 6 were developed and computer-programed. In the late 1980s and in the 1990s (period 3), shortcomings of MPPT became obvious, esp. the sometimes erratic or even divergent behavior of the MPn series. The phys. usefulness of MPPT was questioned and it was suggested to abandon the theory. Since the 1990s (period 4), the focus of method development work has been almost exclusively on MP2. A wealth of techniques and approaches has been put forward to convert MP2 from a O(M5) computational problem into a low-order or linear-scaling task that can handle mols. with thousands of atoms. In addn., the accuracy of MP2 has been systematically improved by introducing spin scaling, dispersion corrections, orbital optimization, or explicit correlation. The coming years will see a continuously strong development in MPPT that will have an essential impact on other quantum chem. methods.**10**Szabados, Á.*Reference Module in Chemistry, Molecular Sciences and Chemical Engineering*; Elsevier, 2017.There is no corresponding record for this reference.**11**Grimme, S. Improved second-order Møller-Plesset perturbation theory by separate scaling of parallel- and antiparallel-spin pair correlation energies.*J. Chem. Phys.*2003,*118*, 9095, DOI: 10.1063/1.156924211https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXjs1Gktbk%253D&md5=4857ad3bcf3e894bb7b94a4d3cf86fc1Improved second-order Moller-Plesset perturbation theory by separate scaling of parallel- and antiparallel-spin pair correlation energiesGrimme, StefanJournal of Chemical Physics (2003), 118 (20), 9095-9102CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A simple modification of the second-order Moller-Plesset perturbation theory (MP2) to improve the description of mol. ground state energies is proposed. The total MP2 correlation energy is partitioned into parallel- and antiparallel-spin components which are sep. scaled. The two parameters (scaling factors), whose values can be justified by basic theor. arguments, were optimized on a benchmark set of 51 reaction energies composed of 74 first-row mols. The new method performs significantly better than std. MP2: the rms [mean abs. error (MAE)] deviation drops from 4.6 (3.3) to 2.3 (1.8) kcal/mol. The max. error is reduced from 13.3 to 5.1 kcal/mol. Significant improvements are esp. obsd. for cases which are usually known as MP2 pitfalls while cases already described well with MP2 remain almost unchanged. Even for 11 atomization energies not considered in the fit, uniform improvements [MAE: 8.1 kcal/mol (MP2) vs. 3.2 kcal/mol (new)] were found. The results are furthermore compared with those from d. functional theory (DFT/B3LYP) and quadratic CI [QCISD/QCISD(T)] calcns. Also for difficult systems including strong (nondynamical) correlation effects, the improved MP2 method clearly outperforms DFT/B3LYP and yields results of QCISD or sometimes QCISD(T) quality. Preliminary calcns. of the equil. bond lengths and harmonic vibrational frequencies for ten diat. mols. also show consistent enhancements. The uniformity with which the new method improves upon MP2, thereby rectifying many of its problems, indicates significant robustness and suggests it as a valuable quantum chem. method of general use.**12**Jung, Y.; Lochan, R. C.; Dutoi, A. D.; Head-Gordon, M. Scaled opposite-spin second order Møller-Plesset correlation energy: An economical electronic structure method.*J. Chem. Phys.*2004,*121*, 9793, DOI: 10.1063/1.180960212https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXpslCitb0%253D&md5=7423ee78fb73040427b9762bfef69705Scaled opposite-spin second order Moller-Plesset correlation energy: An economical electronic structure methodJung, Yousung; Lochan, Rohini C.; Dutoi, Anthony D.; Head-Gordon, MartinJournal of Chemical Physics (2004), 121 (20), 9793-9802CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A simplified approach to treating the electron correlation energy is suggested in which only the α-β component of the second order Moller-Plesset energy is evaluated, and then scaled by an empirical factor which is suggested to be 1.3. This scaled opposite-spin second order energy (SOS-MP2), where MP2 is Moller-Plesset theory, yields results for relative energies and deriv. properties that are statistically improved over the conventional MP2 method. Furthermore, the SOS-MP2 energy can be evaluated without the fifth order computational steps assocd. with MP2 theory, even without exploiting any spatial locality. A fourth order algorithm is given for evaluating the opposite spin MP2 energy using auxiliary basis expansions, and a Laplace approach, and timing comparisons are given.**13**Janesko, B. G.; Scuseria, G. E. Coulomb-only second-order perturbation theory in long-range-corrected hybrid density functionals.*Phys. Chem. Chem. Phys.*2009,*11*, 9677, DOI: 10.1039/b910905f13https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhtlSjtrzF&md5=a246d6986a951f4008845e5adefacdb8Coulomb-only second-order perturbation theory in long-range-corrected hybrid density functionalsJanesko, Benjamin G.; Scuseria, Gustavo E.Physical Chemistry Chemical Physics (2009), 11 (42), 9677-9686CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)We have been investigating the combination of a short-range d. functional approxn. with long-range RPA correlation, where the direct RPA correlation is constructed using only Coulomb (i.e., not antisymmetrized) two-electron integrals. Our group's recently demonstrated connection between RPA and coupled cluster theory suggests investigating a related method: second-order Moller-Plesset perturbation theory correlation (MP2) constructed using only Coulomb integrals. This new "JMP2" method is related to the scaled-opposite-spin SOS-MP2 approxn., which is also constructed using only Coulomb integrals. While JMP2 and SOS-MP2 yield identical results for closed shell systems, they have important differences for open shells. We show here that both JMP2 and SOS-MP2 provide a reasonable treatment of long-range correlation when combined with a short-range exchange-correlation functional. Remarkably, JMP2's explicit inclusion of (approx.) like-spin correlation effects provides significant improvements over SOS-MP2 for thermochem.**14**Szabados, Á.; Nagy, P. Spin component scaling in multiconfiguration perturbation theory.*J. Phys. Chem. A*2011,*115*, 523, DOI: 10.1021/jp108575a14https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXhs1alu77O&md5=b668be917b91b71ddc3d6820c0c55ad1Spin Component Scaling in Multiconfiguration Perturbation TheorySzabados, Agnes; Nagy, PeterJournal of Physical Chemistry A (2011), 115 (4), 523-534CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)We investigate a term-by-term scaling of the second-order energy correction obtained by perturbation theory (PT) starting from a multiconfiguration wave function. The total second-order correction is decompd. into several terms, based on the level and the spin pattern of the excitations. To define individual terms, we extend the same spin/different spin categorization of spin component scaling in various ways. When needed, identification of the excitation level is facilitated by the pivot determinant underlying the multiconfiguration PT framework. Scaling factors are detd. from the stationary condition of the total energy calcd. up to order 3. The decompn. schemes are tested numerically on the example of bond dissocn. profiles and energy differences. We conclude that Grimme's parameters detd. for single-ref. Moller-Plesset theory may give a modest error redn. along the entire potential surface, if adopting a multireference based PT formulation. Scaling factors obtained from the stationary condition show relatively large variation with mol. geometry, at the same time they are more efficient in reducing the error when following a bond dissocn. process.**15**Grimme, S.; Goerigk, L.; Fink, R. F. Spin-component-scaled electron correlation methods.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2012,*2*, 886– 906, DOI: 10.1002/wcms.111015https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXnt1Cksw%253D%253D&md5=94509c70361cb28b2b19b8ef02aee510Spin-component-scaled electron correlation methodsGrimme, Stefan; Goerigk, Lars; Fink, Reinhold F.Wiley Interdisciplinary Reviews: Computational Molecular Science (2012), 2 (6), 886-906CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)A review. Spin-component-scaled (SCS) electron correlation methods for electronic structure theory are reviewed. The methods can be derived theor. by applying special conditions to the underlying wave functions in perturbation theory. They are based on the insight that low-order wave function expansions treat the correlation effects of electron pairs with opposite spin (OS) and same spin (SS) differently because of their different treatment at the underlying Hartree-Fock level. Phys., this is related to the different av. inter-electronic distances in the SS and OS electron pairs. The overview starts with the original SCS-MP2 method and discusses its strengths and weaknesses and various ways to parameterize the scaling factors. Extensions to coupled-cluster and excited state methods as well the connection to virtual-orbital dependent d. functional approaches are highlighted. The performance of various SCS methods in large thermochem. benchmarks and for excitation energies is discussed in comparison with other common electronic structure methods.**16**Grimme, S. Semiempirical hybrid density functional with perturbative second-order correlation.*J. Chem. Phys.*2006,*124*, 034108 DOI: 10.1063/1.214895416https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XptVGnuw%253D%253D&md5=e0e89576e15f6a7c9fb40756b601dc66Semiempirical hybrid density functional with perturbative second-order correlationGrimme, StefanJournal of Chemical Physics (2006), 124 (3), 034108/1-034108/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A new hybrid d. functional for general chem. applications is proposed. It is based on a mixing of std. generalized gradient approxns. (GGAs) for exchange by Becke (B) and for correlation by Lee, Yang, and Parr (LYP) with Hartree-Fock (HF) exchange and a perturbative second-order correlation part (PT2) that is obtained from the Kohn-Sham (GGA) orbitals and eigenvalues. This virtual orbital-dependent functional contains only two global parameters that describe the mixt. of HF and GGA exchange (ax) and of the PT2 and GGA correlation (c), resp. The parameters are obtained in a least-squares-fit procedure to the G2/97 set of heat of formations. Opposed to conventional hybrid functionals, the optimum ax is found to be quite large (53% with c = 27%) which at least in part explains the success for many problematic mol. systems compared to conventional approaches. The performance of the new functional termed B2-PLYP is assessed by the G2/97 std. benchmark set, a second test suite of atoms, mols., and reactions that are considered as electronically very difficult (including transition-metal compds., weakly bonded complexes, and reaction barriers) and comparisons with other hybrid functionals of GGA and meta-GGA types. According to many realistic tests, B2-PLYP can be regarded as the best general purpose d. functional for mols. (e.g., a mean abs. deviation for the two test sets of only 1.8 and 3.2 kcal/mol compared to about 3 and 5 kcal/mol, resp., for the best other d. functionals). Very importantly, also the max. and minium errors (outliers) are strongly reduced (by about 10-20 kcal/mol). Furthermore, very good results are obtained for transition state barriers but unlike previous attempts at such a good description, this definitely comes not at the expense of equil. properties. Preliminary calcns. of the equil. bond lengths and harmonic vibrational frequencies for diat. mols. and transition-metal complexes also show very promising results. The uniformity with which B2-PLYP improves for a wide range of chem. systems emphasizes the need of (virtual) orbital-dependent terms that describe nonlocal electron correlation in accurate exchange-correlation functionals. From a practical point of view, the new functional seems to be very robust and it is thus suggested as an efficient quantum chem. method of general purpose.**17**Sancho-García, J. C.; Adamo, C. Double-hybrid density functionals: Merging wavefunction and density approaches to get the best of both worlds.*Phys. Chem. Chem. Phys.*2013,*15*, 14581, DOI: 10.1039/c3cp50907a17https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXht1KhtbvF&md5=4216f41fe053cdc2348840cbd2567f0cDouble-hybrid density functionals: merging wavefunction and density approaches to get the best of both worldsSancho-Garcia, J. C.; Adamo, C.Physical Chemistry Chemical Physics (2013), 15 (35), 14581-14594CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)A review. We review why and how double-hybrid d. functionals have become new leading actors in the field of computational chem., thanks to the combination of an unprecedented accuracy together with large robustness and reliability. Similar to their predecessors, the widely employed hybrid d. functionals, they are rooted in the Adiabatic Connection Method from which they emerge in a natural way. We present recent achievements concerning applications to chem. systems of the most interest, and current extensions to deal with challenging issues such as non-covalent interactions and excitation energies. These promising methods, despite a slightly higher computational cost than other typical d.-based models, are called to play a key role in the near future and can thus pave the way towards new discoveries or advances.**18**Goerigk, L.; Grimme, S. Double-hybrid density functionals.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2014,*4*, 576, DOI: 10.1002/wcms.119318https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhvVelu7jI&md5=c2c8a4d2d17cea5bc4a9c559d42742c8Double-hybrid density functionalsGoerigk, Lars; Grimme, StefanWiley Interdisciplinary Reviews: Computational Molecular Science (2014), 4 (6), 576-600CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)Double-hybrid d. functionals (DHDFs) are reviewed in this study. In DHDFs parts of conventional d. functional theory (DFT) exchange and correlation are replaced by contributions from nonlocal Fock-exchange and second-order perturbative correlation. The latter portion is based on the well-known MP2 wave-function approach in which, however, Kohn-Sham orbitals are used to calc. its contribution. First, related methods preceding this idea are reviewed, followed by a thorough discussion of the first modern double-hybrid B2-PLYP. Parallels and differences between B2-PLYP and its various successors are then outlined. This discussion is rounded off with representative thermochem. examples demonstrating that DHDFs belong to the most robust and accurate DFT approaches currently available. This anal. also presents hitherto unpublished results for recently developed DHDFs. Finally, how double-hybrids can be combined with linear-response time-dependent DFT is also outlined and the value of this approach for electronically excited states is shown. WIREs Comput Mol Sci 2014, 4:576-600. doi: 10.1002/wcms.1193 For further resources related to this article, please visit the . Conflict of interest: The authors have declared no conflicts of interest for this article.**19**Martin, J. M. L.; Santra, G. Empirical Double-Hybrid Density Functional Theory: A ‘Third Way’ in Between WFT and DFT.*Isr. J. Chem.*2020,*60*, 787, DOI: 10.1002/ijch.20190011419https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXitlyhsrnM&md5=182610d9f5560d261abf36f80a6d9d2eEmpirical Double-Hybrid Density Functional Theory: A 'Third Way' in Between WFT and DFTMartin, Jan M. L.; Santra, GolokeshIsrael Journal of Chemistry (2020), 60 (8-9), 787-804CODEN: ISJCAT; ISSN:0021-2148. (Wiley-VCH Verlag GmbH & Co. KGaA)A review. Double hybrid d. functional theory arguably sits on the seamline between wavefunction methods and DFT: it represents a special case of Rung 5 on the "Jacob's Ladder" of John P. Perdew. For large and chem. diverse benchmarks such as GMTKN55, empirical double hybrid functionals with dispersion corrections can achieve accuracies approaching wavefunction methods at a cost not greatly dissimilar to hybrid DFT approaches, provided RI-MP2 and/or another MP2 acceleration techniques are available in the electronic structure code. Only a half-dozen or fewer empirical parameters are required. For vibrational frequencies, accuracies intermediate between CCSD and CCSD(T) can be achieved, and performance for other properties is encouraging as well. Organometallic reactions can likewise be treated well, provided static correlation is not too strong. Further prospects are discussed, including range-sepd. and RPA-based approaches.**20**Chai, J.-D.; Head-Gordon, M. Long-range corrected double-hybrid density functionals.*J. Chem. Phys.*2009,*131*, 174105 DOI: 10.1063/1.324420920https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhtl2hsLrO&md5=4a5e3b76f67c32897d242da653f7a47cLong-range corrected double-hybrid density functionalsChai, Jeng-Da; Head-Gordon, MartinJournal of Chemical Physics (2009), 131 (17), 174105/1-174105/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We extend the range of applicability of our previous long-range cor. (LC) hybrid functional, ωB97X, with a nonlocal description of electron correlation, inspired by second-order Moller-Plesset (many-body) perturbation theory. This LC "double-hybrid" d. functional, denoted as ωB97X-2, is fully optimized both at the complete basis set limit (using 2-point extrapolation from calcns. using triple and quadruple zeta basis sets), and also sep. using the somewhat less expensive 6-311++G(3df,3pd) basis. On independent test calcns. (as well as training set results), ωB97X-2 yields high accuracy for thermochem., kinetics, and noncovalent interactions. In addn., owing to its high fraction of exact Hartree-Fock exchange, ωB97X-2 shows significant improvement for the systems where self-interaction errors are severe, such as sym. homonuclear radical cations. (c) 2009 American Institute of Physics.**21**Goerigk, L.; Grimme, S. Efficient and Accurate Double-Hybrid-Meta-GGA Density Functionals—Evaluation with the Extended GMTKN30 Database for General Main Group Thermochemistry, Kinetics, and Noncovalent Interactions.*J. Chem. Theory Comput.*2011,*7*, 291, DOI: 10.1021/ct100466k21https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXhs1Srsb7N&md5=bd9fde6f59698f9f9f7a41195e6ad144Efficient and Accurate Double-Hybrid-Meta-GGA Density Functionals-Evaluation with the Extended GMTKN30 Database for General Main Group Thermochemistry, Kinetics, and Noncovalent InteractionsGoerigk, Lars; Grimme, StefanJournal of Chemical Theory and Computation (2011), 7 (2), 291-309CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present an extended and improved version of our recently published database for general main group thermochem., kinetics, and noncovalent interactions, which is dubbed GMTKN30. Furthermore, we suggest and investigate two new double-hybrid-meta-GGA d. functionals called PTPSS-D3 and PWPB95-D3. PTPSS-D3 is based on reparameterized TPSS exchange and correlation contributions; PWPB95-D3 contains reparameterized PW exchange and B95 parts. Both functionals contain fixed amts. of 50% Fock-exchange. Furthermore, they include a spin-opposite scaled perturbative contribution and are combined with our latest atom-pairwise London-dispersion correction. When evaluated with the help of the Laplace transformation algorithm, both methods scale as N4 with system size. The functionals are compared with the double hybrids B2PLYP-D3, B2GPPLYP-D3, DSD-BLYP-D3, and XYG3 for GMTKN30 with a quadruple-ζ basis set. PWPB95-D3 and DSD-BLYP-D3 are the best functionals in our study and turned out to be more robust than B2PLYP-D3 and XYG3. Furthermore, PWPB95-D3 is the least basis set dependent and the best functional at the triple-ζ level. For the example of transition metal carbonyls, it is shown that, mainly due to the lower amt. of Fock-exchange, PWPB95-D3 and PTPSS-D3 are better applicable than the other double hybrids. Finally, we discuss in some detail the XYG3 functional, which makes use of B3LYP orbitals and electron densities. We show that it is basically a highly nonlocal variant of B2PLYP and that its partially good performance is mainly due to a larger effective amt. of perturbative correlation compared to other double hybrids. We finally recommend the PWPB95-D3 functional in general chem. applications.**22**Zhang, I. Y.; Xu, X.; Jung, Y.; Goddard, W. A. A fast doubly hybrid density functional method close to chemical accuracy using a local opposite spin ansatz.*Proc. Natl. Acad. Sci. U.S.A.*2011,*108*, 19896, DOI: 10.1073/pnas.111512310822https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhs12gtbvL&md5=0416b2a69fd6724216a249add03a6385A fast doubly hybrid density functional method close to chemical accuracy using a local opposite spin ansatzZhang, Igor Ying; Xu, Xin; Jung, Yousung; Goddard, William A., IIIProceedings of the National Academy of Sciences of the United States of America (2011), 108 (50), 19896-19900, S19896/1-S19896/10CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)We develop and validate the XYGJ-OS functional, based on the adiabatic connection formalism and Girling-Levy perturbation theory to second order and using the opposite-spin (OS) ansatz combined with locality of electron correlation. XYGJ-OS with local implementation scales as N3 with an overall accuracy of 1.28 kcal/mol for thermochem., bond dissocn. energies, reaction barrier heights, and nonbonded interactions, comparable to that of 1.06 kcal/mol for the accurate coupled-cluster based G3 method (scales as N7) and much better than many popular d. functional theory methods: B3LYP (4.98), PBEO (4.36), and PBE (12.10).**23**Zhang, I. Y.; Su, N. Q.; Brémond, É. A. G.; Adamo, C.; Xu, X. Doubly hybrid density functional xDH-PBE0 from a parameter-free global hybrid model PBE0.*J. Chem. Phys.*2012,*136*, 174103 DOI: 10.1063/1.370389323https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38Xmt12nurw%253D&md5=2f4ea470e5efa8d7da9033f90d27535aDoubly hybrid density functional xDH-PBE0 from a parameter-free global hybrid model PBE0Zhang, Igor Ying; Su, Neil Qiang; Bremond, Eric A. G.; Adamo, Carlo; Xu, XinJournal of Chemical Physics (2012), 136 (17), 174103/1-174103/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Following the XYG3 model which uses orbitals and d. from B3LYP, an empirical doubly hybrid (DH) functional is developed by using inputs from PBE0. This new functional, named xDH-PBE0, has been tested on a no. of different mol. properties, including atomization energies, bond dissocn. enthalpies, reaction barrier heights, and nonbonded interactions. From the results obtained, xDH-PBE0 not only displays a significant improvement with respect to the parent PBE0, but also shows a performance that is comparable to XYG3. Arguably, while PBE0 is a parameter-free global hybrid (GH) functional, the B3LYP GH functional contains eight fit parameters. From a more general point of view, the present work points out that reliable and general-purpose DHs can be obtained with a limited no. of fit parameters. (c) 2012 American Institute of Physics.**24**Kozuch, S.; Martin, J. M. L. Spin-Component-Scaled Double Hybrids: An Extensive Search for the Best Fifth-Rung Functionals Blending DFT and Perturbation Theory.*J. Comput. Chem.*2013,*34*, 2327, DOI: 10.1002/jcc.2339124https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXhtlans7fN&md5=9d34bb3f6cd3fda3be5a41d58ef93075Spin-component-scaled double hybrids: An extensive search for the best fifth-rung functionals blending DFT and perturbation theoryKozuch, Sebastian; Martin, Jan M. L.Journal of Computational Chemistry (2013), 34 (27), 2327-2344CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)Following up on an earlier preliminary communication (Kozuch and Martin, Phys. Chem. Chem. Phys. 2011, 13, 20104), we report here in detail on an extensive search for the most accurate spin-component-scaled double hybrid functionals [of which conventional double hybrids (DHs) are a special case]. Such fifth-rung functionals approach the performance of composite ab initio methods such as G3 theory at a fraction of their computational cost, and with anal. derivs. available. In this article, we provide a crit. anal. of the variables and components that maximize the accuracy of DHs. These include the selection of the exchange and correlation functionals, the coeffs. of each component [d. functional theory (DFT), exact exchange, and perturbative correlation in both the same spin and opposite spin terms], and the addn. of an ad-hoc dispersion correction; we have termed these parametrizations "DSD-DFT" (Dispersion cor., Spin-component scaled, Double-hybrid DFT). Somewhat surprisingly, the quality of DSD-DFT is only mildly dependent on the underlying DFT exchange and correlation components, with even DSD-LDA yielding respectable performance. Simple, nonempirical GGAs appear to work best, whereas meta-GGAs offer no advantage (with the notable exception of B95c). The best correlation components appear to be, in that order, B95c, P86, and PBEc, while essentially any good GGA exchange yields nearly identical results. On further validation with a wider variety of thermochem., weak interaction, kinetic, and spectroscopic benchmarks, we find that the best functionals are, roughly in that order, DSD-PBEhB95, DSD-PBEP86, DSD-PBEPW91, and DSD-PBEPBE. In addn., DSD-PBEP86 and DSD-PBEPBE can be used without source code modifications in a wider variety of electronic structure codes. Sample job decks for several commonly used such codes are supplied as electronic Supporting Information. Copyright © 2013 Wiley Periodicals, Inc.**25**Brémond, É.; Ciofini, I.; Sancho-García, J. C.; Adamo, C. Nonempirical Double-Hybrid Functionals: An Effective Tool for Chemists.*Acc. Chem. Res.*2016,*49*, 1503, DOI: 10.1021/acs.accounts.6b0023225https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xht1ynsb7O&md5=f2722f267174ead8e7ab3e5daf92a76bNonempirical Double-Hybrid Functionals: An Effective Tool for ChemistsBremond, Eric; Ciofini, Ilaria; Sancho-Garcia, Juan Carlos; Adamo, CarloAccounts of Chemical Research (2016), 49 (8), 1503-1513CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)A review. D. functional theory (DFT) emerged in the last two decades as the most reliable tool for the description and prediction of properties of mol. systems and extended materials, coupling in an unprecedented way high accuracy and reasonable computational cost. This success rests also on the development of more and more performing d. functional approxns. (DFAs). Indeed, the Achilles' heel of DFT is represented by the exchange-correlation contribution to the total energy, which, being unknown, must be approximated. Since the beginning of the 1990s, global hybrids (GH) functionals, where an explicit dependence of the exchange-correlation energy on occupied Kohn-Sham orbitals is introduced thanks to a fraction of Hartree-Fock-like exchange, imposed themselves as the most reliable DFAs for chem. applications. However, if these functionals normally provide results of sufficient accuracy for most of the cases analyzed, some properties, such as thermochem. or dispersive interactions, can still be significantly improved. A possible way out is represented by the inclusion, into the exchange-correlation functional, of an explicit dependence on virtual Kohn-Sham orbitals via perturbation theory. This leads to a new class of functionals, called double-hybrids (DHs). In this Account, we describe our nonempirical approach to DHs, which, following the line traced by the Perdew-Burke-Ernzerhof approach, allows for the definition of a GH (PBE0) and a DH (QIDH) model. In such a way, a whole family of nonempirical functionals, spanning on the highest rungs of the Perdew's quality scale, is now available and competitive with other-more empirical-DFAs. Discussion of selected cases, ranging from thermochem. and reactions to weak interactions and excitation energies, not only show the large range of applicability of nonempirical DFAs, but also underline how increasing the no. of theor. constraints parallels with an improvement of the DFA's numerical performances. This fact further consolidates the strong theor. framework of nonempirical DFAs.Finally, even if nonempirical DH approaches are still computationally expensive, relying on the fact that they can benefit of all tech. enhancements developed for speeding up post-Hartree-Fock methods, there is substantial hope for their near future routine application to the description and prediction of complex chem. systems and reactions.**26**Su, N. Q.; Xu, X. The XYG3 Type of Doubly Hybrid Density Functionals.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2016,*6*, 721, DOI: 10.1002/wcms.127426https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhslKgsbfP&md5=be9356f372be073ff2a1794c47a9fc11The XYG3 type of doubly hybrid density functionalsSu, Neil Qiang; Xu, XinWiley Interdisciplinary Reviews: Computational Molecular Science (2016), 6 (6), 721-747CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)Doubly hybrid (DH) functionals have emerged as a new class of d. functional approxns. (DFAs), which not only have a nonlocal orbital-dependent component in the exchange part, but also incorporate the information of unoccupied orbitals in the correlation part, being at the top rung of Perdew's view of Jacob's ladder in DFAs. This review article focuses on the XYG3 type of DH (xDH) functionals, which use a low rung functional to perform the self-consistent-field calcn. to generate orbitals and densities, with which a top rung DH functional is used for final energy evaluation. We will discuss the theor. background of the xDH functionals, briefly reviewing the adiabatic connection formalism, coordinate scaling relations, and Goerling-Levy perturbation theory. General performance of the xDH functionals will be presented for both energies and structures. In particular, we will present the fractional charge behaviors of the xDH functionals, examg. the self-interaction errors, the delocalization errors and the deviation from the linearity condition, as well as their effects on the predicted ionization potentials, electron affinities and fundamental gaps. This provides a theor. rationale for the obsd. good performance of the xDH functionals. WIREs Comput Mol Sci 2016, 6:721-747. doi: 10.1002/wcms.1274 For further resources related to this article, please visit the .**27**Feyereisen, M.; Fitzgerald, G.; Komornicki, A. Use of approximate integrals in ab initio theory. An application in MP2 energy calculations.*Chem. Phys. Lett.*1993,*208*, 359, DOI: 10.1016/0009-2614(93)87156-W27https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXkvVSntL8%253D&md5=4d39ebc6228d81fcd03290405a40dbdbUse of approximate integrals in ab initio theory. An application in MP2 energy calculationsFeyereisen, Martin; Fitzgerald, George; Komornicki, AndrewChemical Physics Letters (1993), 208 (5-6), 359-63CODEN: CHPLBC; ISSN:0009-2614.Authors use the resoln. of the identity (RI) as a convenient way to replace the use of four-index two-electron integrals with linear combinations of three-index integrals. The method is broadly applicable to a wide range of problems in quantum chem. Authors demonstrate the effectiveness of RI for the calcn. of MP2 energies. For the water dimer, agreement within 0.1 kcal/mol is obtained with respect to exact MP2 calcns. The RI-MP2 energies require only about 10% of the time required by conventional MP2.**28**Gyevi-Nagy, L.; Kállay, M.; Nagy, P. R. Integral-direct and parallel implementation of the CCSD(T) method: Algorithmic developments and large-scale applications.*J. Chem. Theory Comput.*2020,*16*, 366, DOI: 10.1021/acs.jctc.9b0095728https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BB3MfjvFShsw%253D%253D&md5=62e3430fb75945f63928d737c0cb0d49Integral-Direct and Parallel Implementation of the CCSD(T) Method: Algorithmic Developments and Large-Scale ApplicationsGyevi-Nagy Laszlo; Kallay Mihaly; Nagy Peter RJournal of chemical theory and computation (2020), 16 (1), 366-384 ISSN:.A completely integral-direct, disk I/O, and network traffic economic coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] implementation has been developed relying on the density-fitting approximation. By fully exploiting the permutational symmetry, the presented algorithm is highly operation count and memory-efficient. Our measurements demonstrate excellent strong scaling achieved via hybrid MPI/OpenMP parallelization and a highly competitive, 60-70% utilization of the theoretical peak performance on up to hundreds of cores. The terms whose evaluation time becomes significant only for small- to medium-sized examples have also been extensively optimized. Consequently, high performance is also expected for systems appearing in extensive data sets used, e.g., for density functional or machine learning parametrizations, and in calculations required for certain reduced-cost or local approximations of CCSD(T), such as in our local natural orbital scheme [LNO-CCSD(T)]. The efficiency of this implementation allowed us to perform some of the largest CCSD(T) calculations ever presented for systems of 31-43 atoms and 1037-1569 orbitals using only four to eight many-core CPUs and 1-3 days of wall time. The resulting 13 correlation energies and the 12 corresponding reaction energies and barrier heights are added to our previous benchmark set collecting reference CCSD(T) results of molecules at the applicability limit of current implementations.**29**Almlöf, J. Elimination of energy denominators in Møller-Plesset perturbation theory by a Laplace transform approach.*Chem. Phys. Lett.*1991,*181*, 319, DOI: 10.1016/0009-2614(91)80078-C29https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3MXkvF2msL8%253D&md5=7b12bd4b2091509f65775ab277389b0eElimination of energy denominators in Moeller-Plesset perturbation theory by a Laplace transform approachAlmlof, JanChemical Physics Letters (1991), 181 (4), 319-20CODEN: CHPLBC; ISSN:0009-2614.It is shown how the energy denominators encountered in various schemes for electronic structure calcn. can be removed by a Laplace transform technique. The method is applicable to a wide variety of electronic structure calcns.**30**Häser, M.; Almlöf, J. Laplace transform techniques in Møller-Plesset perturbation theory.*J. Chem. Phys.*1992,*96*, 489, DOI: 10.1063/1.462485There is no corresponding record for this reference.**31**Ayala, P. Y.; Scuseria, G. E. Linear scaling second-order Møller-Plesset theory in the atomic orbital basis for large molecular systems.*J. Chem. Phys.*1999,*110*, 3660, DOI: 10.1063/1.47825631https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXosVWmsQ%253D%253D&md5=1d3a71f19d43f89ec50a315a1d96db18Linear scaling second-order Moeller-Plesset theory in the atomic orbital basis for large molecular systemsAyala, Philippe Y.; Scuseria, Gustavo E.Journal of Chemical Physics (1999), 110 (8), 3660-3671CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We have used Almlof and Haser's Laplace transform idea to eliminate the energy denominator in second-order perturbation theory (MP2) and obtain an energy expression in the AO basis. We show that the asymptotic computational cost of this method scales quadratically with mol. size. We then define AO domains such that selective pairwise interactions can be neglected using well-defined thresholding criteria based on the power law decay properties of the long-range contributions. For large mols., our scheme yields linear scaling computational cost as a function of mol. size. The errors can be controlled in a precise manner and our method reproduces canonical MP2 energies. We present benchmark calcns. of polyglycine chains and water clusters contg. up to 3040 basis functions.**32**Surján, P. R. The MP2 energy as a functional of the Hartree-Fock density matrix.*Chem. Phys. Lett.*2005,*406*, 318– 320, DOI: 10.1016/j.cplett.2005.03.02432https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXjtF2gsbw%253D&md5=375a4f31cb04f375d948e95c7a03cdfbThe MP2 energy as a functional of the Hartree-Fock density matrixSurjan, Peter R.Chemical Physics Letters (2005), 406 (4-6), 318-320CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.)The explicit E[2][P] functional is presented, where E [2] is the second order Moller-Plesset correlation energy and P is the std. Hartree-Fock d. matrix. The ideas leading to this functional are implicit in previous studies, but the significance of its existence has not yet been sufficiently emphasized and its simple explicit form has not been presented. With the proposed functional one may obtain the correlation energy in the absence of MOs, knowing merely the d. matrix. This may further facilitate linear scaling computation of the correlation energy.**33**Kobayashi, M.; Nakai, H. Implementation of Surján’s density matrix formulae for calculating second-order Møller-Plesset energy.*Chem. Phys. Lett.*2006,*420*, 250– 255, DOI: 10.1016/j.cplett.2005.12.08833https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28Xhslyqtrg%253D&md5=89e912a120e1e1d4d979b725e710f2c2Implementation of Surjan's density matrix formulae for calculating second-order Moller-Plesset energyKobayashi, Masato; Nakai, HiromiChemical Physics Letters (2006), 420 (1-3), 250-255CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.)We numerically assess the method for obtaining second-order Moller-Plesset (MP2) energy from the Hartree-Fock d. matrix (DM) recently proposed by Surjan [Surjan, Chem. Phys. Lett. 406 (2005) 318]. It is confirmed that Surjan's method, referred to as DM-Laplace MP2, can obtain MP2 energy accurately by means of appropriate integral quadrature and a matrix exponential evaluation scheme. Numerical tests reveal that the Euler-Maclaurin and the Romberg numerical integration schemes can achieve milli-hartree accuracy with small quadrature points. This Letter also indicates the possibility of the application of DM-Laplace MP2 to linear-scaling SCF techniques, which give approx. DM.**34**Doser, B.; Lambrecht, D. S.; Kussmann, J.; Ochsenfeld, C. Linear-scaling atomic orbital-based second-order Møller-Plesset perturbation theory by rigorous integral screening criteria.*J. Chem. Phys.*2009,*130*, 064107 DOI: 10.1063/1.307290334https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhvFentbY%253D&md5=5814416cce1e9f2d0e3140008c61358dLinear-scaling atomic orbital-based second-order Moller-Plesset perturbation theory by rigorous integral screening criteriaDoser, Bernd; Lambrecht, Daniel S.; Kussmann, Joerg; Ochsenfeld, ChristianJournal of Chemical Physics (2009), 130 (6), 064107/1-064107/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A Laplace-transformed second-order Moller-Plesset perturbation theory (MP2) method is presented, which allows to achieve linear scaling of the computational effort with mol. size for electronically local structures. Also for systems with a delocalized electronic structure, a cubic or even quadratic scaling behavior is achieved. Numerically significant contributions to the AO (AO)-MP2 energy are preselected using the so-called multipole-based integral ests. (MBIE) introduced earlier by us. Since MBIE provides rigorous upper bounds, numerical accuracy is fully controlled and the exact MP2 result is attained. While the choice of thresholds for a specific accuracy is only weakly dependent upon the mol. system, our AO-MP2 scheme offers the possibility for incremental thresholding: for only little addnl. computational expense, the numerical accuracy can be systematically converged. We illustrate this dependence upon numerical thresholds for the calcn. of intermol. interaction energies for the S22 test set. The efficiency and accuracy of our AO-MP2 method is demonstrated for linear alkanes, stacked DNA base pairs, and carbon nanotubes: e.g., for DNA systems the crossover toward conventional MP2 schemes occurs between one and two base pairs. In this way, it is for the first time possible to compute wave function-based correlation energies for systems contg. more than 1000 atoms with 10 000 basis functions as illustrated for a 16 base pair DNA system on a single-core computer, where no empirical restrictions are introduced and numerical accuracy is fully preserved. (c) 2009 American Institute of Physics.**35**Schäfer, T.; Ramberger, B.; Kresse, G. Quartic scaling MP2 for solids: A highly parallelized algorithm in the plane wave basis.*J. Chem. Phys.*2017,*146*, 104101 DOI: 10.1063/1.497693735https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC1czms1Knsw%253D%253D&md5=054a349b32abeee80c9412e2aeccf107Quartic scaling MP2 for solids: A highly parallelized algorithm in the plane wave basisSchafer Tobias; Ramberger Benjamin; Kresse GeorgThe Journal of chemical physics (2017), 146 (10), 104101 ISSN:.We present a low-complexity algorithm to calculate the correlation energy of periodic systems in second-order Moller-Plesset (MP2) perturbation theory. In contrast to previous approximation-free MP2 codes, our implementation possesses a quartic scaling, O(N(4)), with respect to the system size N and offers an almost ideal parallelization efficiency. The general issue that the correlation energy converges slowly with the number of basis functions is eased by an internal basis set extrapolation. The key concept to reduce the scaling is to eliminate all summations over virtual orbitals which can be elegantly achieved in the Laplace transformed MP2 formulation using plane wave basis sets and fast Fourier transforms. Analogously, this approach could allow us to calculate second order screened exchange as well as particle-hole ladder diagrams with a similar low complexity. Hence, the presented method can be considered as a step towards systematically improved correlation energies.**36**Pulay, P.; Saebø, S. Orbital-invariant formulation and second-order gradient evaluation in Møller-Plesset perturbation theory.*Theor. Chem. Acc.*1986,*69*, 357, DOI: 10.1007/BF0052669736https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL28XltVakt7g%253D&md5=d16c8d33f0c76e557d97ae63f8cfca22Orbital-invariant formulation and second-order gradient evaluation in Moeller-Plesset perturbation theoryPulay, Peter; Saeboe, SveinTheoretica Chimica Acta (1986), 69 (5-6), 357-68CODEN: TCHAAM; ISSN:0040-5744.Based on the Hylleraas functional form, the second and third orders of the Moeller-Plesset (MP) perturbation theory were reformulated in terms of arbitrary (e.g., localized) internal orbitals, and AOs in the virtual space. The results are strictly equiv. to the canonical formulation if no further approxns. are introduced. The new formalism permits the extension of the local correlation method to MP theory. It also facilitates the treatment of weak pairs at a lower (e.g., second-order) level of theory in CI and coupled-cluster methods. Based on the formalism, an MP2 gradient algorithm is outlined, which does not require the storage of deriv. integrals, integrals with three external MO indexes, and, using the method of N. C. Handy and H. F. Schaefer III (1984), the repeated soln. of the coupled-perturbed SCF equations.**37**Kats, D.; Usvyat, D.; Schütz, M. On the use of the Laplace transform in local correlation methods.*Phys. Chem. Chem. Phys.*2008,*10*, 3430, DOI: 10.1039/b802993h37https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXmvVemtLw%253D&md5=5dd80442ae7cda8caa0d66efe739483fOn the use of the Laplace transform in local correlation methodsKats, Danylo; Usvyat, Denis; Schuetz, MartinPhysical Chemistry Chemical Physics (2008), 10 (23), 3430-3439CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)The applicability of the Laplace transform ansatz of Almlof in the context of local correlation methods with a priori restricted sets of wavefunction parameters is explored. A new local MP2 method based on the Laplace transform ansatz is described, its relation to the local MP2 method based on the Pulay ansatz is elucidated, and its accuracy and efficiency are compared to the latter.**38**Nagy, P. R.; Samu, G.; Kállay, M. An integral-direct linear-scaling second-order Møller-Plesset approach.*J. Chem. Theory Comput.*2016,*12*, 4897, DOI: 10.1021/acs.jctc.6b0073238https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhsV2lu7zO&md5=abd0db52ed97c1a4246e976e14b0aca1An Integral-Direct Linear-Scaling Second-Order Moller-Plesset ApproachNagy, Peter R.; Samu, Gyula; Kallay, MihalyJournal of Chemical Theory and Computation (2016), 12 (10), 4897-4914CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)An integral-direct, iteration-free, linear-scaling, local second-order Moller-Plesset (MP2) approach is presented, which is also useful for spin-scaled MP2 calcns. as well as for the efficient evaluation of the perturbative terms of double-hybrid d. functionals. The method is based on a fragmentation approxn.: the correlation contributions of the individual electron pairs are evaluated in domains constructed for the corresponding localized orbitals, and the correlation energies of distant electron pairs are computed with multipole expansions. The required electron repulsion integrals are calcd. directly invoking the d. fitting approxn.; the storage of integrals and intermediates is avoided. The approach also utilizes natural auxiliary functions to reduce the size of the auxiliary basis of the domains and thereby the operation count and memory requirement. Our test calcns. show that the approach recovers 99.9% of the canonical MP2 correlation energy and reproduces reaction energies with an av. (max.) error below 1 kJ/mol (4 kJ/mol). Our benchmark calcns. demonstrate that the new method enables MP2 calcns. for mols. with more than 2300 atoms and 26000 basis functions on a single processor.**39**Saebø, S.*Linear-Scaling Techniques in Computational Chemistry and Physics: Methods and Applications*; Springer: Netherlands, 2011; pp 65– 82.There is no corresponding record for this reference.**40**Zienau, J.; Clin, L.; Doser, B.; Ochsenfeld, C. Cholesky-decomposed densities in Laplace-based second-order Møller-Plesset perturbation theory.*J. Chem. Phys.*2009,*130*, 204112 DOI: 10.1063/1.314259240https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXntVSgur0%253D&md5=34625433d59701160948f2dff3305a2aCholesky-decomposed densities in Laplace-based second-order Moller-Plesset perturbation theoryZienau, Jan; Clin, Lucien; Doser, Bernd; Ochsenfeld, ChristianJournal of Chemical Physics (2009), 130 (20), 204112/1-204112/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Based on our linear-scaling AO second-order Moller-Plesset perturbation theory (AO-MP2) method , we explore the use of Cholesky-decompd. pseudodensity (CDD) matrixes within the Laplace formulation. Numerically significant contributions are preselected using our multipole-based integral ests. as upper bounds to two-electron integrals so that the 1/R6 decay behavior of transformed Coulomb-type products is exploited. In addn., we combine our new CDD-MP2 method with the resoln. of the identity (RI) approach. Even though the use of RI results in a method that shows a quadratic scaling behavior in the dominant steps, gains of up to one or two orders of magnitude vs. our original AO-MP2 method are obsd. in particular for larger basis sets. (c) 2009 American Institute of Physics.**41**Maurer, S. A.; Clin, L.; Ochsenfeld, C. Cholesky-decomposed density MP2 with density fitting: Accurate MP2 and double-hybrid DFT energies for large systems.*J. Chem. Phys.*2014,*140*, 224112 DOI: 10.1063/1.488114441https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXpslCqu7c%253D&md5=9463d1755513305cadf4f01aa16cd2cbCholesky-decomposed density MP2 with density fitting: Accurate MP2 and double-hybrid DFT energies for large systemsMaurer, Simon A.; Clin, Lucien; Ochsenfeld, ChristianJournal of Chemical Physics (2014), 140 (22), 224112/1-224112/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Our recently developed QQR-type integral screening is introduced in our Cholesky-decompd. pseudo-densities Moller-Plesset perturbation theory of second order (CDD-MP2) method. We use the resoln.-of-the-identity (RI) approxn. in combination with efficient integral transformations employing sparse matrix multiplications. The RI-CDD-MP2 method shows an asymptotic cubic scaling behavior with system size and a small prefactor that results in an early crossover to conventional methods for both small and large basis sets. We also explore the use of local fitting approxns. which allow to further reduce the scaling behavior for very large systems. The reliability of our method is demonstrated on test sets for interaction and reaction energies of medium sized systems and on a diverse selection from our own benchmark set for total energies of larger systems. Timings on DNA systems show that fast calcns. for systems with more than 500 atoms are feasible using a single processor core. Parallelization extends the range of accessible system sizes on one computing node with multiple cores to more than 1000 atoms in a double-zeta basis and more than 500 atoms in a triple-zeta basis. (c) 2014 American Institute of Physics.**42**Helmich-Paris, B.; Repisky, M.; Visscher, L. Relativistic Cholesky-decomposed density matrix MP2.*Chem. Phys.*2019,*518*, 38, DOI: 10.1016/j.chemphys.2018.11.00942https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXitlenu7jN&md5=4eeed0769da6e3f6709e0e313aea7514Relativistic Cholesky-decomposed density matrix MP2Helmich-Paris, Benjamin; Repisky, Michal; Visscher, LucasChemical Physics (2019), 518 (), 38-46CODEN: CMPHC2; ISSN:0301-0104. (Elsevier B.V.)We introduce the relativistic Cholesky-decompd. d. (CDD) matrix second-order Moller-Plesset perturbation theory (MP2) energies. The working equations are formulated in terms of the usual intermediates of MP2 when employing the resoln.-of-the-identity approxn. (RI) for two-electron integrals. Those intermediates are obtained by substituting the occupied and virtual quaternion pseudo-d. matrixes of our previously proposed two-component (2C) AO-based MP2 (Helmich-Paris et al., 2016) by the corresponding pivoted quaternion Cholesky factors. While working within the Kramers-restricted formalism, we obtain a formal spin-orbit overhead of 16 and 28 for the Coulomb and exchange contribution to the 2C MP2 correlation energy, resp., compared to a non-relativistic (NR) spin-free CDD-MP2 implementation. This compact quaternion formulation could also be easily explored in any other algorithm to compute the 2C MP2 energy. The quaternion Cholesky factors become sparse for large mols. and, with a block-wise screening, block sparse-matrix multiplication algorithm, we obsd. an effective quadratic scaling of the total wall time for heavy-element contg. linear mols. with increasing system size. The total run time for both NR and 2C calcns. was dominated by the contraction to the exchange energy. We have also investigated a bulky Te-contg. supramol. complex. For such bulky, three-dimensionally extended mols. the present screening scheme has a much larger prefactor and is less effective.**43**Glasbrenner, M.; Graf, D.; Ochsenfeld, C. Efficient Reduced-Scaling Second-Order Møller-Plesset Perturbation Theory with Cholesky-Decomposed Densities and an Attenuated Coulomb Metric.*J. Chem. Theory Comput.*2020,*16*, 6856, DOI: 10.1021/acs.jctc.0c0060043https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXitVykurrM&md5=e1b5deb7744946700ddc855d1c69f960Efficient Reduced-Scaling Second-Order Moller-Plesset Perturbation Theory with Cholesky-Decomposed Densities and an Attenuated Coulomb MetricGlasbrenner, Michael; Graf, Daniel; Ochsenfeld, ChristianJournal of Chemical Theory and Computation (2020), 16 (11), 6856-6868CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a novel, highly efficient method for the computation of second-order Moller-Plesset perturbation theory (MP2) correlation energies, which uses the resoln. of the identity (RI) approxn. and local MOs obtained from a Cholesky decompn. of pseudodensity matrixes (CDD), as in the RI-CDD-MP2 method developed previously in our group [Maurer, S.A. et al., J. Chem. Phys., 2014, 140, 224112]. In addn., we introduce an attenuated Coulomb metric and subsequently redesign the RI-CDD-MP2 method in order to exploit the resulting sparsity in the three-center integrals. Coulomb and exchange energy contributions are computed sep. using specialized algorithms. A simple, yet effective integral screening protocol based on Schwarz ests. is used for the MP2 exchange energy. The Coulomb energy computation and the preceding transformations of the three-center integrals are accelerated using a modified version of the natural blocking approach [Jung, Y., Head-Gordon, M., Phys. Chem. Chem. Phys., 2006, 8, 2831]. Effective subquadratic scaling for a wide range of mol. sizes is demonstrated in test calcns. in conjunction with a low prefactor. The method is shown to enable cost-efficient MP2 calcns. on large mol. systems with several thousand basis functions.**44**Neuhauser, D.; Rabani, E.; Baer, R. Expeditious Stochastic Approach for MP2 Energies in Large Electronic Systems.*J. Chem. Theory Comput.*2013,*9*, 24, DOI: 10.1021/ct300946j44https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhvVSqtL7E&md5=00873c7c04cdecfa46df550c78ee1cf7Expeditious Stochastic Approach for MP2 Energies in Large Electronic SystemsNeuhauser, Daniel; Rabani, Eran; Baer, RoiJournal of Chemical Theory and Computation (2013), 9 (1), 24-27CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A fast stochastic method for calcg. the second order Moller-Plesset (MP2) correction to the correlation energy of large systems of electrons is presented. The approach is based on reducing the exact summation over occupied and unoccupied states to a time-dependent trace formula amenable to stochastic sampling. We demonstrate the abilities of the method to treat systems with thousands of electrons using hydrogen passivated silicon spherical nanocrystals represented on a real space grid, much beyond the capabilities of present day MP2 implementations.**45**Willow, S. Y.; Kim, K. S.; Hirata, S. Stochastic evaluation of second-order many-body perturbation energies.*J. Chem. Phys.*2012,*137*, 204122 DOI: 10.1063/1.476869745https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhslKms77I&md5=2c542905be62e0d3ab11a74fb3e4786aStochastic evaluation of second-order many-body perturbation energiesWillow, Soohaeng Yoo; Kim, Kwang S.; Hirata, SoJournal of Chemical Physics (2012), 137 (20), 204122/1-204122/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)With the aid of the Laplace transform, the canonical expression of the second-order many-body perturbation correction to electronic energy is converted into a sum of two 13-dimensional integrals, the 12-dimensional parts of which are evaluated by Monte Carlo integration. Wt. functions are identified that are anal. normalizable, finite, non-neg. everywhere, and share the same singularities as the integrands. They generate appropriate distributions of four-electron walkers via the Metropolis algorithm, yielding correlation energies of small mols. within a few mEh of the correct values after 108 Monte Carlo steps. This algorithm does away with the integral transformation as the hotspot of the usual algorithms, has a far superior size dependence of cost, does not suffer from the sign problem of some quantum Monte Carlo methods, and potentially easily parallelizable and extensible to other more complex electron-correlation theories. (c) 2012 American Institute of Physics.**46**Barca, G. M. J.; McKenzie, S. C.; Bloomfield, N. J.; Gilbert, A. T. B.; Gill, P. M. W. Q-MP2-OS: Møller-Plesset Correlation Energy by Quadrature.*J. Chem. Theory Comput.*2020,*16*, 1568, DOI: 10.1021/acs.jctc.9b0114246https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhsFKhtrs%253D&md5=5bc60925dc5e5142669a39d0141d02e8Q-MP2-OS: Moller-Plesset Correlation Energy by QuadratureBarca, Giuseppe M. J.; McKenzie, Simon C.; Bloomfield, Nathaniel J.; Gilbert, Andrew T. B.; Gill, Peter M. W.Journal of Chemical Theory and Computation (2020), 16 (3), 1568-1577CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a quadrature-based algorithm for computing the opposite-spin component of the MP2 correlation energy which scales quadratically with basis set size and is well-suited to large-scale parallelization. The key ideas, which are rooted in the earlier work of Hirata and co-workers, are to abandon all two-electron integrals, recast the energy as a seven-dimensional integral, approx. that integral by quadrature, and employ a cutoff strategy to minimize the no. of intermediate quantities. We discuss our implementation in detail and show that it parallelizes almost perfectly on 840 cores for cyclosporine (a mol. with roughly 200 atoms), exhibits scaling for a sequence of polyglycines, and is principally limited by the accuracy of its quadrature.**47**Martínez, T. J.; Carter, E. A. Pseudospectral Møller-Plesset perturbation theory through third order.*J. Chem. Phys.*1994,*100*, 3631, DOI: 10.1063/1.46635047https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXit1Onsrc%253D&md5=54542779da7d2ff27e0d4038bcf4d056Pseudospectral Moeller-Plesset perturbation theory through third orderMartinez, Todd J.; Carter, Emily A.Journal of Chemical Physics (1994), 100 (5), 3631-8CODEN: JCPSA6; ISSN:0021-9606.The authors present a formulation and implementation of Moeller-Plesset perturbation theory in a pseudospectral framework. At the second-order level, the pseudospectral formulation is a formally a factor of N/n faster than conventional approaches, while the third order is formally faster by a factor of n, where N is the no. of AOs and n is the no. of occupied orbitals. The accuracy of the resulting energies is probed for a no. of test cases. Practical timings are presented and show conclusively that the pseudospectral formulation is faster than conventional ones.**48**Kossmann, S.; Neese, F. Efficient Structure Optimization with Second-Order Many-Body Perturbation Theory: The RIJCOSX-MP2 Method.*J. Chem. Theory Comput.*2010,*6*, 2325, DOI: 10.1021/ct100199k48https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXot1GisLk%253D&md5=6dd45c3962123232a7e9e5fd36630f8aEfficient Structure Optimization with Second-Order Many-Body Perturbation Theory: The RIJCOSX-MP2 MethodKossmann, Simone; Neese, FrankJournal of Chemical Theory and Computation (2010), 6 (8), 2325-2338CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Efficient energy calcns. and structure optimizations employing second-order Moller-Plesset perturbation theory (MP2) are presented. The application of the RIJCOSX approxn., which involves different approxns. for the formation of the Coulomb- and exchange-type matrixes, to MP2 theory is demonstrated. The RIJCOSX approxn. incorporates the resoln. of the identity' approxn. in terms of a Split-RI-J variant for the evaluation of the Coulomb matrixes and a seminumeric exchange treatment via the chain-of-spheres' algorithm for the formation of the exchange-type matrixes. Beside the derivation of the working equations, the RIJCOSX-MP2 method is benchmarked against the original MP2 and the already highly efficient RI-MP2 method. Energies as well as gradients are computed employing various basis sets and are compared to the conventional MP2 results concerning accuracy and total wall clock times. Speedups of typically a factor of 5-7 in comparison to MP2 can be obsd. for the largest basis set employed in our study. Total energies are reproduced with an av. error of ≤0.8 kcal/mol and min. energy geometries differ by ∼0.1 pm in bond lengths and typically ∼0.2 degrees in bond angles. The RIJCOSX-MP2 gradient parallelizes with a speedup of 8.2 on 10 processors. The algorithms are implemented into the ORCA electronic structure package.**49**Maslen, P. E.; Head-Gordon, M. Non-iterative local second order Møller-Plesset theory.*Chem. Phys. Lett.*1998,*283*, 102, DOI: 10.1016/S0009-2614(97)01333-X49https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXhtlCqs7Y%253D&md5=999d3729e8ab38176986d00ffeee0634Non-iterative local second order Moller-Plesset theoryMaslen, P. E.; Head-Gordon, M.Chemical Physics Letters (1998), 283 (1,2), 102-108CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)Second order Moller-Plesset perturbation theory (MP2) is formulated in terms of atom-centered occupied and virtual orbitals. Both the occupied and the virtual orbitals are non-orthogonal. A new parameter-free atoms-in-mols. local approxn. is employed to reduce the cost of the calcn. to cubic scaling, and a quasi-canonical two-particle basis is introduced to enable the soln. of the local MP2 equations via explicit matrix diagonalization rather than iteration.**50**Jung, Y.; Shao, Y.; Head-Gordon, M. Fast evaluation of scaled opposite-spin second-order Møller-Plesset correlation energies using auxiliary basis expansions and exploiting sparsity.*J. Comput. Chem.*2007,*28*, 1953, DOI: 10.1002/jcc.2059050https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXotVynsbk%253D&md5=50644e7112ac163100d2e7daca8e0d99Fast evaluation of scaled opposite spin second-order Moller-Plesset correlation energies using auxiliary basis expansions and exploiting sparsityJung, Yousung; Shao, Yihan; Head-Gordon, MartinJournal of Computational Chemistry (2007), 28 (12), 1953-1964CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)The scaled opposite spin Moller-Plesset method (SOS-MP2) is an economical way of obtaining correlation energies that are computationally cheaper, and yet, in a statistical sense, of higher quality than std. MP2 theory, by introducing one empirical parameter. But SOS-MP2 still has a fourth-order scaling step that makes the method inapplicable to very large mol. systems. We reduce the scaling of SOS-MP2 by exploiting the sparsity of expansion coeffs. and local integral matrixes, by performing local auxiliary basis expansions for the occupied-virtual product distributions. To exploit sparsity of 3-index local quantities, we use a blocking scheme in which entire zero-rows and columns, for a given third global index, are deleted by comparison against a numerical threshold. This approach minimizes sparse matrix book-keeping overhead, and also provides sufficiently large submatrixes after blocking, to allow efficient matrix-matrix multiplies. The resulting algorithm is formally cubic scaling, and requires only moderate computational resources (quadratic memory and disk space) and, in favorable cases, is shown to yield effective quadratic scaling behavior in the size regime we can apply it to. Errors assocd. with local fitting using the attenuated Coulomb metric and numerical thresholds in the blocking procedure are found to be insignificant in terms of the predicted relative energies. A diverse set of test calcns. shows that the size of system where significant computational savings can be achieved depends strongly on the dimensionality of the system, and the extent of localizability of the MOs.**51**Förster, A.; Franchini, M.; van Lenthe, E.; Visscher, L. A Quadratic Pair Atomic Resolution of the Identity Based SOS-AO-MP2 Algorithm Using Slater Type Orbitals.*J. Chem. Theory Comput.*2020,*16*, 875, DOI: 10.1021/acs.jctc.9b0085451https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BB3MbpsVCgsA%253D%253D&md5=f19110433d706d7ce8a95981b55a3573A Quadratic Pair Atomic Resolution of the Identity Based SOS-AO-MP2 Algorithm Using Slater Type OrbitalsForster Arno; Franchini Mirko; Visscher Lucas; Franchini Mirko; van Lenthe ErikJournal of chemical theory and computation (2020), 16 (2), 875-891 ISSN:.We report a production level implementation of pair atomic resolution of the identity (PARI) based second-order Moller-Plesset perturbation theory (MP2) in the Slater type orbital (STO) based Amsterdam Density Functional (ADF) code. As demonstrated by systematic benchmarks, dimerization and isomerization energies obtained with our code using STO basis sets of triple-ζ-quality show mean absolute deviations from Gaussian type orbital, canonical, basis set limit extrapolated, global density fitting (DF)-MP2 results of less than 1 kcal/mol. Furthermore, we introduce a quadratic scaling atomic orbital based spin-opposite-scaled (SOS)-MP2 approach with a very small prefactor. Due to a worst-case scaling of [Formula: see text], our implementation is very fast already for small systems and shows an exceptionally early crossover to canonical SOS-PARI-MP2. We report computational wall time results for linear as well as for realistic three-dimensional molecules and show that triple-ζ quality calculations on molecules of several hundreds of atoms are only a matter of a few hours on a single compute node, the bottleneck of the computations being the SCF rather than the post-SCF energy correction.**52**Förster, A.; Visscher, L. Double hybrid DFT calculations with Slater type orbitals.*J. Comput. Chem.*2020,*41*, 1660, DOI: 10.1002/jcc.2620952https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BB38zlvFGrtw%253D%253D&md5=df9dd9cc70a8109012fa4f17204467f6Double hybrid DFT calculations with Slater type orbitalsForster Arno; Visscher LucasJournal of computational chemistry (2020), 41 (18), 1660-1684 ISSN:.On a comprehensive database with 1,644 datapoints, covering several aspects of main-group as well as of transition metal chemistry, we assess the performance of 60 density functional approximations (DFA), among them 36 double hybrids (DH). All calculations are performed using a Slater type orbital (STO) basis set of triple-ζ (TZ) quality and the highly efficient pair atomic resolution of the identity approach for the exchange- and Coulomb-term of the KS matrix (PARI-K and PARI-J, respectively) and for the evaluation of the MP2 energy correction (PARI-MP2). Employing the quadratic scaling SOS-AO-PARI-MP2 algorithm, DHs based on the spin-opposite-scaled (SOS) MP2 approximation are benchmarked against a database of large molecules. We evaluate the accuracy of STO/PARI calculations for B3LYP as well as for the DH B2GP-PLYP and show that the combined basis set and PARI-error is comparable to the one obtained using the well-known def2-TZVPP Gaussian-type basis set in conjunction with global density fitting. While quadruple-ζ (QZ) calculations are currently not feasible for PARI-MP2 due to numerical issues, we show that, on the TZ level, Jacob's ladder for classifying DFAs is reproduced. However, while the best DHs are more accurate than the best hybrids, the improvements are less pronounced than the ones commonly found on the QZ level. For conformers of organic molecules and noncovalent interactions where very high accuracy is required for qualitatively correct results, DHs provide only small improvements over hybrids, while they still excel in thermochemistry, kinetics, transition metal chemistry and the description of strained organic systems.**53**Hohenstein, E. G.; Parrish, R. M.; Martínez, T. J. Tensor hypercontraction density fitting. I. Quartic scaling second- and third-order Møller-Plesset perturbation theory.*J. Chem. Phys.*2012,*137*, 044103 DOI: 10.1063/1.473231053https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhtVOgtb7M&md5=5ce6ce5cd9f7915a7d02ed5bc5fed4f2Tensor hypercontraction density fitting. I. Quartic scaling second- and third-order Moller-Plesset perturbation theoryHohenstein, Edward G.; Parrish, Robert M.; Martinez, Todd J.Journal of Chemical Physics (2012), 137 (4), 044103/1-044103/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Many approxns. have been developed to help deal with the O(N4) growth of the electron repulsion integral (ERI) tensor, where N is the no. of one-electron basis functions used to represent the electronic wavefunction. Of these, the d. fitting (DF) approxn. is currently the most widely used despite the fact that it is often incapable of altering the underlying scaling of computational effort with respect to mol. size. We present a method for exploiting sparsity in three-center overlap integrals through tensor decompn. to obtain a low-rank approxn. to d. fitting (tensor hypercontraction d. fitting or THC-DF). This new approxn. reduces the 4th-order ERI tensor to a product of five matrixes, simultaneously reducing the storage requirement as well as increasing the flexibility to regroup terms and reduce scaling behavior. As an example, we demonstrate such a scaling redn. for second- and third-order perturbation theory (MP2 and MP3), showing that both can be carried out in O(N4) operations. This should be compared to the usual scaling behavior of O(N5) and O(N6) for MP2 and MP3, resp. The THC-DF technique can also be applied to other methods in electronic structure theory, such as coupled-cluster and CI, promising significant gains in computational efficiency and storage redn. (c) 2012 American Institute of Physics.**54**Bangerter, F. H.; Glasbrenner, M.; Ochsenfeld, C. Low-Scaling Tensor Hypercontraction in the Cholesky Molecular Orbital Basis Applied to Second-Order Møller-Plesset Perturbation Theory.*J. Chem. Theory Comput.*2021,*17*, 211, DOI: 10.1021/acs.jctc.0c0093454https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXis12rtLrN&md5=2a77224fa989a3f704c8aca79c12d584Low-Scaling Tensor Hypercontraction in the Cholesky Molecular Orbital Basis Applied to Second-Order Moller-Plesset Perturbation TheoryBangerter, Felix H.; Glasbrenner, Michael; Ochsenfeld, ChristianJournal of Chemical Theory and Computation (2021), 17 (1), 211-221CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We employ various reduced scaling techniques to accelerate the recently developed least-squares tensor hypercontraction (LS-THC) approxn. [Parrish, R.M. et al., J. Chem. Phys. 2012, 137, 224106] for electron repulsion integrals (ERIs) and apply it to second-order Moller-Plesset perturbation theory (MP2). The grid-projected ERI tensors are efficiently constructed using a localized Cholesky MO basis from d.-fitted integrals with an attenuated Coulomb metric. Addnl., rigorous integral screening and the natural blocking matrix format are applied to reduce the complexity of this step. By recasting the equations to form the quantized representation of the 1/r operator Z into the form of a system of linear equations, the bottleneck of inverting the grid metric via pseudoinversion is removed. This leads to a reduced scaling THC algorithm and application to MP2 yields the (sub-)quadratically scaling THC-ω-RI-CDD-SOS-MP2 method. The efficiency of this method is assessed for various systems including DNA fragments with over 8000 basis functions and the subquadratic scaling is illustrated.**55**Del Ben, M.; Hutter, J.; VandeVondele, J. Second-Order Møller-Plesset Perturbation Theory in the Condensed Phase: An Efficient and Massively Parallel Gaussian and Plane Waves Approach.*J. Chem. Theory Comput.*2012,*8*, 4177, DOI: 10.1021/ct300531w55https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38Xht12isL3F&md5=90469d4482aae04ee25d1fe5de6e8621Second-Order Moller-Plesset Perturbation Theory in the Condensed Phase: An Efficient and Massively Parallel Gaussian and Plane Waves ApproachDel Ben, Mauro; Hutter, Jurg; VandeVondele, JoostJournal of Chemical Theory and Computation (2012), 8 (11), 4177-4188CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A novel algorithm, based on a hybrid Gaussian and plane waves (GPW) approach, is developed for the canonical second-order Moller-Plesset perturbation energy (MP2) of finite and extended systems. The key aspect of the method is that the electron repulsion integrals (ia|λσ) are computed by direct integration between the products of Gaussian basis functions λσ and the electrostatic potential arising from a given occupied-virtual pair d. ia. The electrostatic potential is obtained in a plane waves basis set after solving the Poisson equation in Fourier space. In particular, for condensed phase systems, this scheme is highly efficient. Furthermore, our implementation has low memory requirements and displays excellent parallel scalability up to 100 000 processes. In this way, canonical MP2 calcns. for condensed phase systems contg. hundreds of atoms or more than 5000 basis functions can be performed within minutes, while systems up to 1000 atoms and 10 000 basis functions remain feasible. Solid LiH has been employed as a benchmark to study basis set and system size convergence. Lattice consts. and cohesive energies of various mol. crystals have been studied with MP2 and double-hybrid functionals.**56**Katouda, M.; Naruse, A.; Hirano, Y.; Nakajima, T. Massively parallel algorithm and implementation of RI-MP2 energy calculation for peta-scale many-core supercomputers.*J. Comput. Chem.*2016,*37*, 2623, DOI: 10.1002/jcc.2449156Massively parallel algorithm and implementation of RI-MP2 energy calculation for peta-scale many-core supercomputersKatouda, Michio; Naruse, Akira; Hirano, Yukihiko; Nakajima, TakahitoJournal of Computational Chemistry (2016), 37 (30), 2623-2633CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)A new parallel algorithm and its implementation for the RI-MP2 energy calcn. utilizing peta-flop-class many-core supercomputers are presented. Some improvements from the previous algorithm have been performed: (1) a dual-level hierarchical parallelization scheme that enables the use of more than 10,000 Message Passing Interface (MPI) processes and (2) a new data communication scheme that reduces network communication overhead. A multi-node and multi-GPU implementation of the present algorithm is presented for calcns. on a central processing unit (CPU)/graphics processing unit (GPU) hybrid supercomputer. Benchmark results of the new algorithm and its implementation using the K computer (CPU clustering system) and TSUBAME 2.5 (CPU/GPU hybrid system) demonstrate high efficiency. The peak performance of 3.1 PFLOPS is attained using 80,199 nodes of the K computer. The peak performance of the multi-node and multi-GPU implementation is 514 TFLOPS using 1349